# PointCrack3D: Crack Detection in Unstructured Environments using a 3D-Point-
Cloud-Based Deep Neural Network∗

Faris Azhari1,2, Charlotte Sennersten3, Michael Milford1 and Thierry Peynot1,2
*This work was supported by Australian Government Research Training Program
(RTP), Queensland University of Technology (QUT) through the QUT Centre for
Robotics and Mining3.1QUT Centre for Robotics, Brisbane, Australia.
<EMAIL_ADDRESS>Pinjarra Hills, Australia.3CSIRO
Mineral Resources, Pullenvale, Australia.

###### Abstract

Surface cracks on buildings, natural walls and underground mine tunnels can
indicate serious structural integrity issues that threaten the safety of the
structure and people in the environment. Timely detection and monitoring of
cracks is crucial to managing these risks, especially if the systems can be
made highly automated through robots. Vision-based crack detection algorithms
using deep neural networks have exhibited promise for structured surfaces such
as walls or civil engineering tunnels, but little work has addressed highly
unstructured environments such as rock cliffs and bare mining tunnels. To
address this challenge, this paper presents PointCrack3D, a new 3D-point-
cloud-based crack detection algorithm for unstructured surfaces. The method
comprises three key components: an adaptive down-sampling method that
maintains sufficient crack point density, a DNN that classifies each point as
crack or non-crack, and a post-processing clustering method that groups crack
points into crack _instances_. The method was validated experimentally on a
new large natural rock dataset, comprising coloured LIDAR point clouds
spanning more than 900 m2 and 412 individual cracks. Results demonstrate a
crack detection rate of 97% overall and 100% for cracks with a maximum width
of more than 3 cm, significantly outperforming the state of the art.
Furthermore, for cross-validation, PointCrack3D was applied to an entirely new
dataset acquired in different location and not used at all in training and
shown to detect 100% of its crack instances. We also characterise the
relationship between detection performance, crack width and number of points
per crack, providing a foundation upon which to make decisions about both
practical deployments and future research directions.

##  I Introduction

Cracks on structures, whether human-made (e.g. pavements, buildings, bridges,
tunnels) or natural (e.g. cliffs, rock surfaces, caves), can be defined as the
structure that breaks a surface continuity with a gap. They are signs of
potential weakness, and can ultimately lead to catastrophic disasters such as
collapses and rockfalls. For example, in an underground mining environment,
surface cracks (also known as fractures) are one of the early signs of
geological failures that can lead to rockfalls or even collapse of the roof
[1, 2]. Over a 10-year period (2010 - 2019), in the U.S.A. alone, 49 deaths
and 3,359 non-fatal accidents were reported as a result of geological failures
[3]. This toll accounts for over 25% of the total death toll in underground
mines. Current mitigation techniques include routine visual inspections that
pose a risk to the safety of the geotechnical engineer, especially when
surveying voids that are abandoned, newly blasted or post-ore extraction.
Automated inspection systems using robots could eliminate this risk and
increase survey frequency, resulting in earlier detection. This work focuses
on the development of a key component of this system: automated crack
detection. This detector should be capable of: detecting crack _instances_ (so
they can be counted), determining their position (globally and relatively to
each other) and characterising them (e.g. estimating their width and length).

Figure 1: Scanning an unstructured surface with cracks using a 3D LIDAR
scanner with the coloured point cloud and labelled crack (red) overlaid.

A large body of research has investigated the detection of cracks in
structured environments, on mostly flat or smooth surface, using vision e.g.
[4]. However, as noted in [5], in imagery of an unstructured surface, shadows
of a protruding structure can easily be confused with an actual crack. Prior
work has shown that the performance of state-of-the-art (SOTA) image-based
crack detection methods is significantly compromised when applied to a
shotcrete-layered unstructured surface in an underground mine site [5], even
though the surfaces in this case remain relatively smooth thanks to the
shotcrete. In this paper we aim to detect cracks in significantly more
challenging scenes, such as highly unstructured bare rock surfaces, see Fig. 1
and Fig. 3.

To address the challenge, the automated crack detection method proposed in
this paper exploits LIDAR data to capture accurate geometric information. The
properties of most cracks - a thin opening that covers a relatively small area
on the surface - makes for a challenging problem as point clouds in the crack
region can be sparser compared to neighbouring flat contiguous surfaces.
Inspired by promising early results obtained using DNN methods, this paper
proposes PointCrack3D (PtCrack3D), a point-cloud-based DNN approach for crack
detection on highly unstructured surfaces. PtCrack3D comprises three key
components: an adaptive down-sampling method that maintains sufficient crack
point density, a DNN that classifies each point as crack or non-crack, and a
post-processing clustering method that groups crack points into crack
_instances_.

The proposed method was trained and validated experimentally on a new dataset
of a large natural rock cliff, on which it is shown to detect 97% of the crack
instances overall, and 100% of the cracks at least 3 cm wide, thereby strongly
outperforming a state-of-the-art point-cloud based defects detection method.
Furthermore, the paper proposes a cross-validation of the method, where
PtCrack3D applied ‘out of the box’ is shown to detect 100% of the crack
instances of an entirely new dataset acquired in different location and not
used at all in the training.

The rest of the paper is structured as follows. Sec. II discusses the relevant
literature. Sec. III outlines the crack detection method and implementation
followed by Sec. V, which evaluates the performance of the approach. Finally,
conclusions and future work are discussed in Sec. VI.

##  II Related Work

The automation of crack detection has been a topic of wide interest that
follows along with the advancement and affordability of perception sensors
such as cameras and LIDARs. In addition, with the advancement of Deep Neural
Network (DNN) methods, utilising DNN architectures has become the SOTA in
detecting cracks [4, 6, 7, 8]. However, existing research has focused on
detecting cracks on structured surface i.e. mostly flat or with a known
geometry such as on pavements [4, 7, 8], concrete [6, 7, 4] or tunnels [9]
using monocular vision. These vision-based methods have shown promising
results, but have been seen to struggle when applied to more challenging
environments where the surface is unstructured [5]. 3D sensors have also been
utilised for crack and defects detection, however, they have been applied on
similar structural domains such as pavements [10, 11], concrete [12, 13, 14],
timber [15] and aeroplane exterior [16].

To the best of the authors’ knowledge no prior work achieved LIDAR-based crack
detection method in an unstructured environment using deep learning. Hence, in
this section, relevant literature discussed concerns:  (a) point cloud-based
defects detection systems;  (b) DNN architecture for point cloud.

###  II-A Point cloud-based defects detection methods

Point cloud-based crack detection has not been explored as much as its image-
based counterpart. Hence, this section explores defects detection, a more
general application domain. In [12, 14], the point clouds of concrete surfaces
are projected to a 2D plane producing range images and are used for detection,
for example, applying DNN method on the images [17]. Other methods include
grid search [10] and plane fitting [15] which have been applied on pavement
and timber point clouds, respectively. Defects on aeroplane exteriors have
been detected using region growing on the surface normal estimation [16]. In
[18], Model Point Feature Histogram (MPFH) uses local surface descriptors
paired with a machine learning method to detect defects on a variety of point
cloud surfaces (e.g. welding, artificial teeth, ceramics) where one of the
defects are surface cracks. MPFH was reported to achieve an average accuracy
of 94% in the best configuration. In these applications, the surfaces are flat
or at least a priori known and could be easily modelled, unlike the highly
unstructured surfaces in this work.

###  II-B Deep neural network architecture for point cloud

PointNet [19] was the first DNN method that performs convolution directly on
the raw point cloud input without the need of any point cloud representation
such as voxelisation or projection. Further works based on PointNet was done
such as adding local region computation [20, 21]. Based on local region
computations, [22, 23] added a module that computes the correlation between
the extracted local regions and achieved better performance compared to
methods without it. Additionally, implementations such as [24, 25, 26] were
proposed for fast large-scale point cloud segmentation. However, these methods
were never tested on large imbalanced datasets in a unstructured environment
where the object of interest is small compared to the scene and represented by
points in low density.

Point cloud-based DNN methods have been successfully applied for semantic
segmentation on real-world data, for example, segmenting ground, buildings and
vegetation of aerial point cloud [27], indoor and outdoor object segmentation
in [28, 29], respectively. However, point cloud DNN has not been applied to
segmentation problems with high class imbalance.

##  III PointCrack3D

This paper proposes PointCrack3D, a point-cloud-based crack detection
algorithm for unstructured surfaces using a DNN architecture. Given a point
cloud of an unstructured surface, the objective is first to classify each
point into either crack or non-crack and then to cluster crack points into
their respective crack instances to perform crack instance detection. The
system consists of three components:  (a) Data preparation;  (b) DNN
architecture;  (c) Point clustering.

###  III-A Pre-Processing

Given a point cloud of a scanned surface, the raw data is first discretised
into voxels. The point clouds contained in each voxel are then sampled down to
meet the required DNN input resolution followed by normalisation of the point
coordinates and its corresponding feature set.

The data preparation process is governed by the parameter $[d,n,s]$, where $d$
is the dimension of the voxel, $n$ is the number of points inside a voxel and
$s$ is the stride at which the voxel moves in space, where $s=d$ means no
overlapping between neighbouring voxels. Each voxel that exceeds $n$ number of
points is then down-sampled using an adapted voxel grid-based method as shown
in Algorithm 1. Respective to the nature of cracks points on a surface point
cloud, this down-sampling method preserves the sparse points that represent a
crack compared to applying random sampling method which leads to the
diminishing of sparse crack points. This discretisation step (i.e.
voxelisation followed by down-sampling) preserves the geometrical features of
cracks on a large surface compared to using a whole down-sampled point cloud
directly as an input.

Finally the voxelised points and their respective features are normalised
within the range of [0,1] with respect to the entire training dataset which
finally gives the input as:

$\textstyle V_{i=1...v_{n}}=\textstyle P_{j}\\{X_{j},F_{j}\\}_{j=1...n}$ (1)

where $v_{n}$ is the number of voxels, $X_{j}$ and $F_{j}$ are the normalised
coordinate, $\\{x_{j},y_{j},z_{j}\\}$ and the normalised features of the
$j$-th points, and $n$ is the number of points in a voxel. This gives an input
size of $n\times m$ where $m$ depends on the dimension of features included,
$m=3+dim(F)$.

Input: Point cloud = $pc$, No. desired points = $n$

Output: Down-sampled $pc=pc_{out}$

$grid=floor(\sqrt[3]{n})$;

$v=voxelise(pc,grid)$;

while  _$size(v) <n$_ do

$grid\mathrel{{+}{+}}$;

$v=voxelise(pc,grid)$;

end while

if  _$size(v) >n$_ then

$v_{s}=$ choose $[size(v)-n]$ randomly from $v$;

foreach  _$i\in index(v_{s})$_ do

$v_{n}$ = choose a random voxel neighbour of $[v_{s}(i)|\notin v_{s}]$;

$v_{s}(i)=[v_{s}(i)\ ;\ v_{n}]$;

end foreach

end if

foreach  _$i\in index(v)$_ do

$c=mean(v_{pc}(i))$;

$pc_{out}(i)=min(dist(v_{pc}(i),c))$;

end foreach

Algorithm 1 Modified voxel grid sampling

###  III-B Deep neural network architecture

Figure 2: Illustration of PointConv architecture used. Convolution:
(32,0.1,2048) is a PointConv layer with neighbourhood $sigma=32$, $radius=0.1$
and $centroids=2048$. Deconvolution: (16,0.1) is a PointDeConv layer with
neighbourhood $sigma=32$ and $radius=0.1$. Dense(128,0.5) is a fully connected
layer with output dimension of 128 followed by a dropout layer with a rate of
0.5.

The DNN architecture based on PointConv [22] was adapted for the segmentation
of crack points because it allows for point-by-point segmentation, and learns
directly from input points without the need of converting to another form of
representation and computes local region correlation. The architectural layers
used is as shown in Fig. 2.

Given a dataset $S=\\{V_{i},G_{i}\\}$ where $V_{i}$ is the $i$-th voxel
containing $n$ data points, $P_{j=1...n}$, and $G_{i}$ is the binary ground
truth of size $n\times 1$, {0,1} corresponding to the class label {non-
crack,crack} of each data points. The confidence level, $h(P_{j}|z_{j})$ as a
crack point is given by the sigmoid function:

$\textstyle h(P_{j}|z_{j})=\frac{1}{1+exp(-z_{j})}$ (2)

where $z_{j}$ is the output corresponding to $P_{j}$ from the final dense
layer of the architecture. $h(P_{j}|z_{j})$ has a value [0,1]. To generalise,
the notation $H_{\ast}$, i.e. $H_{i}$ for the confidence level of each point
in the $i^{th}$ voxel and $H_{S}$ for dataset $S$, will be used throughout
this paper to denote the output from the DNN architecture given the input
dataset $S$.

###  III-C Point Clustering

The output from the DNN only tells us how confident the prediction is on each
point in a particular voxel but does not make any inference on whether the
points are an instance of a single crack. Grouping points into clusters
belonging to the same instance of crack enables the extraction of measurements
of the cluster of points i.e. crack such as length, width, or volume.

The proposed post-processing follows these steps:  (1) Reconstruct surface
point clouds from voxels.  (2) For each surface point cloud, take points $p$
with $H_{p}\geq\Delta_{H}$ (i.e. points with confidence level greater or equal
to $\Delta_{H}$), where $\Delta_{H}$ is a pre-defined threshold. Points with
$H_{p}<\Delta_{H}$ are labelled as non-crack.  (3) The points in $p$ are
grouped into clusters, where two points are considered to be from the same
cluster if their distance is less than a set threshold $\Delta_{r}$.  (4) For
each cluster on a surface, reject clusters that have fewer points than a set
threshold $\Delta_{n}$ and assign the points as non-cracks. For the remaining
clusters, assign a similar value to each point of a cluster, with distinct
values across clusters.

##  IV Implementation

This section describes how PtCrack3D was validated which includes the nature
of the dataset, hyperparameters, hardware and the metrics used to evaluate the
performance of the proposed crack detection system.

###  IV-A Dataset

To date, there is no publicly available dataset collected for the purpose of
evaluating crack detection on an unstructured surface, hence, a new data
collection was collected. The Kangaroo Point cliffs (about 18 m high) located
in Brisbane, Australia was chosen as the surfaces are unstructured and contain
many cracks. A FARO® Focus3D x330 HDR LIDAR scanner was used to scan the
cliffs section-by-section (between 15 to 30 m each). The scanning setting of
the LIDAR scanner was set at a constant value of a resolution of 20,480
vertical points per one complete resolution and a scan quality of 6/8. This
decision was made due to the scanner’s capability (i.e. would not allow maxing
out both quality and resolution) based on visually inspecting and deciding the
balance needed for the point cloud to be able to capture the geometry of
cracks. In total there are 21 scans that include the RGB values and the
intensity of the reflected beam for each point. Each point was then labelled
by hand into two categories: crack and non-crack, hence, a point $j$ contains
$P_{j}\\{x_{j},y_{j},z_{j},F_{j}\\{r_{j},g_{j},b_{j},i_{j}\\}\\{g_{i}\\}$
where $\\{x,y,z\\}$ are the position relative to the sensor, $\\{r,g,b,i\\}$
are the RGB information and intensity respectively, while $g_{i}$ is the added
labelled ground truth.

In total, there are 412 cracks across KP dataset. Two-thirds of the cracks are
randomly assigned for training while the rest are for testing. For the
training set, surface points that are within 15 cm from the edges of a crack
are included as negative. The restriction on the number of surface points
included increases the crack points representation to 10%. The training set is
then split randomly into training and validation with a ratio of 2:1. The
remaining surface points are included in the testing set. In the testing set,
crack points represent 0.08% of the total points.

The point clouds are then voxelised, $voxel_{d,n,s}$ using $d=0.5~{}m$ and
$n=2048$. These numbers are chosen based on experimentation where it gives
favourable results while balancing the appropriate density of voxels. As for
$s$, different values were tested, $s=[d,0.35\times d,0.30\times d,0.25\times
d]$. For the testing set, $s$ is set equal to $d$. Voxels with less than $n$
points are discarded from each set. The system was also tested on varying
input feature combination. During training, each input point in each voxel is
perturbed by $0.001\times r$ m limited to the range $[-0.005,0.005]$ where $r$
is a value randomly sampled from a standard normal distribution.

Figure 3: Far Left: Cross section samples of the unstructured surface. Left:
Output of PtCrack3D with $\Delta_{H}=0.59$. Coloured point cloud of a surface
with TP (blue), FN (red) and FP (cyan) layered on top. Right: A true crack
detection zoomed in. Far Right: A false crack detection zoomed in.

###  IV-B Hyperparamaters

The hyperparameters used for the PointConv architecture are as shown in Table
I. Additionally, the bias of the last layer i.e. Dense(1), of the DNN
architecture (fig. 2) is initialised to $log(N_{pos}/N_{neg})$ where $N_{pos}$
and $N_{neg}$ is the number of crack points and number of non-crack points in
the training dataset respectively. As for the focal loss parameters, all
possible combinations of $\gamma$ and $\alpha$ values were tested.

TABLE I: Parameters configuration for the pointConv architecture. Hyperparameters | Value
---|---
Optimiser | Adam ($\beta_{1/2}=(0.9,0.999)$, $\epsilon=10e^{-7}$)
Epochs | 101
Learning rate, $lr$ | 0.01
Learning rate decay |  $lr=0.5\times lr$ after every 10 epochs
Loss function | focal loss = ($\gamma,\alpha$) [30]
Focal Loss parameters | ([2,3,4,5],[10,25,50,75,90])
Batch normalisation | true
Batch size | 5

###  IV-C Evaluation metrics

The performance of the output from the DNN segmentation method is evaluated in
terms of the per-point precision, recall and specificity score defined
respectively as:

$\begin{gathered}\textstyle precision=\frac{TP}{TP+FP}\text{ , }\textstyle
recall=\frac{TP}{TP+FN}\\\ \textstyle
specificity=\frac{TN}{TN+FP}\end{gathered}$ (3)

where true positives ($TP$) are the number of actual crack points classified
as crack, false negatives ($FN$) are the number of actual crack points
classified as non-cracks and false positives ($FP$) are the number of surface
points classified as cracks. These metrics give the performance of the
detector at the per-point level, however, it does not capture the performance
of the detector at per-crack level (crack instance detection). Hence, we
define the rate of crack instances detected, $cr_{det}$ as:

$cr_{det}=\frac{1}{N_{cr}}\sum_{n=1}^{N_{cr}}\begin{cases}1,&\text{if any
}cr_{pred}\cap cr_{real}\\\ 0,&\text{otherwise}\end{cases}$ (4)

where $N_{cr}$ is the total number of cracks in the dataset, $cr_{pred}$ is an
instance of detected crack and $cr_{real}$ is an instance of a real crack. In
Eq. 4, the condition $cr_{pred}\cap cr_{real}$ (i.e. a detected crack instance
intersects a real crack instance) is only true if the {number of intersected
points} $\geq$ {$\alpha*$number of points in $cr_{pred}$}, where $\alpha$ is a
number between (0,1]. It is also true if one or more $cr_{pred}$ intersects
one $cr_{real}$, hence this metric is paired with crack continuity:

$cr_{con}=\textstyle\frac{1}{N_{cr}}\sum_{n=1}^{N_{cr}}\frac{1}{N_{n,cr_{pred}\cap
cr_{real}}}$ (5)

where $N_{cr}$ is the total number of real cracks in the dataset and
$N_{n,cr_{pred}\cap cr_{real}}$ is the number of predicted crack instance that
satisfies the condition $cr_{pred}\cap cr_{real}$ for the $n^{th}$ crack. This
metric measures the fragmentation of the detection on a crack. Finally crack
precision, $cr_{pre}$ is defined as:

$cr_{pre}=\textstyle\frac{N_{cr_{pred}\cap cr_{real}}}{N_{cr_{pred}}}$ (6)

where $N_{cr_{pred}\cap cr_{real}}$ is the number of predicted crack instance
that satisfies the condition $cr_{pred}\cap cr_{real}$ and $N_{cr_{pred}}$ is
the total number of predicted crack instances in the dataset. This metric
gives the ratio of crack clusters detected to the total clusters detected.
Each metrics generates a value between [0,1] where a value closer to one
indicates better performance. Note that these metrics can only be evaluated
after the post-processing step as the output from this step are clusters of
points where each cluster is considered as an instance of detected cracks.

###  IV-D Training Execution

PtCrack3D was trained and tested on an Intel i7-8700K, 64GB RAM with NVIDIA
GeForce GTX 1080 Ti graphics card. The training set used was augmented by
translating the point cloud 10 times along randomly picked axes by a value
randomly generated from within [0, 0.50] m. The results shown in this section
are based on the validation set. The goal of the experimentation was to
determine the best DNN configuration and parameters for the crack detection
system.

The first test was to discover the best $\gamma$ and $\alpha$ values for the
focal loss function. For this test, the position $(x,y,z)$ of the point was
used as the only feature input. While a number of combinations were used (as
in Table I), only tests showing promising results were included in Table II.
The configuration with the highest F1 score was chosen. To further justify the
use of focal loss and the chosen parameters, the focal loss result was
compared to the result using binary cross-entropy as shown in Table II – last
row. The F1 score and recall value of the focal loss was higher than of the
binary cross-entropy, however, binary cross-entropy performed better in terms
of precision which is common when there is a class imbalance towards negative
samples. By applying focal loss, during training, less weight is given to easy
sample points i.e. non-crack and vice versa. For the application of crack
detection where failure to detect cracks could lead to disastrous
consequences, a higher recall value is more desirable, whilst compromising the
precision.

TABLE II: Point-wise performance with varying focal loss parameters. $\gamma$ |  $\alpha$ | Precision | Recall | F1 score
---|---|---|---|---
3 |  0.75 | 0.32 | 0.23 | 0.27
3 |  0.90 | 0.25 | 0.68 | 0.37
4 |  0.75 | 0.39 | 0.41 | 0.40
4 |  0.90 | 0.24 | 0.66 | 0.35
5 |  0.75 | 0.32 | 0.26 | 0.29
5 |  0.90 | 0.22 | 0.64 | 0.33
Binary cross entropy | 0.64 | 0.21 | 0.31

We evaluated the effect of using different combination of point features, see
Table III. The results show that adding colour information ($rgb$) on top of
position-only ($xyz$) increases the performance in both precision and recall
by about 10% and 20% respectively. This is because the structure of cracks
often makes them appear darker than their surrounding surface. Adding the
LIDAR intensity $i$ to $xyz$ increases performance by almost 5%, however,
adding $i$ on top of $rgb$ does not seem to alter the performance as it is
likely that $rgb$ already captures the information useful in discriminating
cracks.

TABLE III: Point-wise detection results with varying feature input. Feature input, s (m) | Precision | Recall | F1 score
---|---|---|---
$x,y,z$ | 0.39 | 0.41 | 0.40
$x,y,z,i$ | 0.43 | 0.45 | 0.44
$x,y,z,r,g,b$ | 0.50 | 0.60 | 0.55
$x,y,z,r,g,b,i$ | 0.50 | 0.60 | 0.55

Finally we tested the impact of varying stride lengths using the original
training set i.e. without the translation augmentation, see Table IV.
Increasing the overlapping increases the performance in all aspects. This is
due to the network having more chance of learning a certain feature multiple
times in different positions in a voxel. However, having lower stride length
increases the number of input voxels, hence increasing the time required to
complete the training of 101 epochs. As in Table IV – 3 last rows, an increase
of 5% overlap caused the training duration to almost double.

TABLE IV: Point-wise detection performance with varying voxel stride. voxel stride, s (m) [% overlap] | Precision | Recall | F1 score |  Duration (d,h,m)
---|---|---|---|---
0.500 [0%] | 0.45 | 0.47 | 0.46 |  0,02,29
0.250 [50%] | 0.50 | 0.50 | 0.50 |  0,15,37
0.175 [65%] | 0.54 | 0.60 | 0.57 |  1,10,38
0.150 [70%] | 0.58 | 0.63 | 0.60 |  2,07,45
0.125 [75%] | 0.60 | 0.67 | 0.63 |  4,02,31

##  V Experimental Results

The detection performance of PtCrack3D was evaluated using the model trained
with voxel stride of 0.125 m, and compared to SOTA defects detection, MPFH
[18] as mentioned in Sec. II. For a fair comparison, MPFH was fine-tuned to
our dataset. For our MPFH implementation, a neural network was used after the
feature extraction step. The $rgb$ values of each point were also included as
a feature on top of the extracted ones, adding 3 additional features for the
neural net. This addition of RGB values increased the overall performance of
MPFH with an absolute increase of around 7% for precision and recall during
training.

###  V-A Pointwise Classification

Fig. 4 shows PtCrack3D per-point classification performance (in solid lines).
At threshold, $\Delta_{H}=0.5$, PtCrack3D obtained a recall score of 76%,
specificity of 92% and a precision of 0.7%. The low precision score is due to
the large class imbalance where small percentage of surface points falsely
detected as crack points will exponentially decrease the precision score.

In comparison, MPFH, shown in dotted-lines in Fig. 4 at $\Delta_{H}=0.5$,
achieved a recall score of 8%, which is 68% lower than PtCrack3D in absolute.
The precision is consistent at 0.5% across all threshold values while ours
scored 13% maximum when the threshold is $\Delta_{H}=0.74$.

Figure 4: Performance curves of PtCrack3D on Kangaroo Point (KP) and Coolum
Beach (CB) dataset and MPFH applied on KP.

###  V-B Crack Detection

This section focuses on the detection of crack instances, after the post-
processing step of PtCrack3D. Table V shows the results for different
confidence thresholds and post-processing configurations. For our method, we
found that $\Delta_{H}=0.59$ provides the most relevant balance between
precision and recall, given the application. In this configuration the method
detected 97% of the cracks in the test dataset, even though only half of the
crack points were classified as belonging to a crack. For MPFH, the results
are shown for $\Delta_{H}=0.005$ and $[\Delta_{r}=0.04,\Delta_{n}=15]$ where
the performance was best (both precision and recall are at their maximum).
MPFH was only able to detect 47% of the cracks instances. PtCrack3D clearly
outperforms MPFH, even with the addition of colour, across all metrics.
Increasing the threshold $\Delta_{H}$ to 0.65 for our method, leads to a
similar recall to that obtained with MPFH, but with a much higher precision
and still a significantly higher crack detection rate of 78%.

Figure 5: Crack detection output in relation to crack width and number of crack points for each individual crack in the test set (dot is the mean width while the ends are the width extremities). Left: PtCrack3D. Right: MPFH. TABLE V: Performance of crack detection methods. $\Delta_{H}$ = Confidence threshold, [$\Delta_{r}$, $\Delta_{n}$] = [distance threshold, point threshold]. | PtCrack3D |  MPFH
---|---|---
$\Delta_{H}$ | 0.50 | 0.59 | 0.65 | 0.005
[$\Delta_{r}$, $\Delta_{n}$] | [0.02,20] | [0.04,20] | [0.04,15] | [0.04,15]
precision | 0.013 | 0.05 | 0.15 | 0.023
recall | 0.73 | 0.49 | 0.26 | 0.29
specificity | 0.95 | 0.99 | 0.99 | 0.99
$cr_{det}$ | 0.99 | 0.97 | 0.78 | 0.47
$cr_{con}$ | 0.81 | 0.92 | 0.84 | 0.83
$cr_{pre}$ | 0.0075 | 0.027 | 0.084 | 0.02

Further evaluation of our method (with $\Delta_{H}=0.59$) was performed by
analysing the detection outcome against the width and the number of points of
each crack instance, see Fig. 5. The width of a crack was measured by taking
the shortest distances between the points on the opposite edges of the crack
at a constant interval across its length. This was done manually from the
labelled point cloud. Fig. 5 shows that the proposed method successfully
detected 100% of the cracks with a maximum width of $0.03\text{\,}\mathrm{m}$
or above. In contrast, MPFH detected 91% of the 11 cracks with a maximum width
above $0.06\text{\,}\mathrm{m}$ and only 61% of the 57 cracks with a maximum
width above $0.03\text{\,}\mathrm{m}$.

PtCrack3D was able to detect 100% of crack with more than 500 points, which is
similar to MPFH. As for cracks with 500 points and below, PtCrack3D managed to
detect 97% of them while MPFH only detected 40%. Having more points for a
small crack allows for more relevant information to be extracted compared to
small cracks with fewer points.

At $\Delta_{H}=0.65$ and $0.005$ for PtCrack3D and MPFH, where the per-point
recall scores are almost similar, PtCrack3D was able to detect more cracks
compared to MPFH. This is because our method is more consistent in the ability
to detect crack points across each crack (i.e. being able to detect about the
same percentage of points across all cracks) while MPFH detects more points on
easier cracks (i.e. cracks with larger width or points) and failed to detect
any points on harder cases.

Qualitative analysis of the results shows that our system was able to detect
cracks with sufficient points for characterisation. As shown in Fig. 3, where
the coloured points i.e. blue, red and cyan are TP, FN and FP crack points
classified by the method. The FN are scattered in groups of small numbers
where the blue clusters are sufficient to extract crack characteristics such
as length. However, PtCrack3D also has the tendency to detect sharp-concaved
edges as a crack as can be seen in numbers on the right-hand side of the full
surface in the left-most image of Fig. 3 with a close-up view of a sample on
the left-most image.

###  V-C Cross-validation

A second dataset was collected to test the trained model on a different scene.
Scans of unstructured rock surfaces of Coolum Beach (see Fig. 1), located
$120\text{\,}\mathrm{k}\mathrm{m}$ north of Kangaroo Point, were captured
using the same LIDAR scanner and configuration. The rock formations are
between $1.5\text{\,}\mathrm{-}$$3\text{\,}\mathrm{m}$ in height and
$3\text{\,}\mathrm{-}$$5\text{\,}\mathrm{m}$ in width which allows for high
density point cloud across the surfaces. In total, 5 scans were captured
covering a total surface area of approximately more than
$100\text{\,}\mathrm{m}^{2}$ with 35 cracks representing about 7% of the total
points with width ranging from $0.5\text{\,}\mathrm{c}\mathrm{m}$ to
$35\text{\,}\mathrm{c}\mathrm{m}$.

####  V-C1 Implementation

PtCrack3D was applied on the entire Coolum dataset without any re-training,
with the same implementation and parameters as in Sec. IV except for the post-
processing step. The $[\Delta_{r},\Delta_{n}]$ parameters of the point
clustering were set to $[0.05,300]$ to account for the larger dimensions of
the cracks and higher point density due to the sensor being closer to the
scanned surface.

####  V-C2 Results and Discussions

PtCrack3D managed to detect all of the cracks in the dataset in terms of
crack-wise performance. Note that the smallest crack is about
$0.5\text{\,}\mathrm{c}\mathrm{m}$ and $0.7\text{\,}\mathrm{c}\mathrm{m}$ in
width at the smallest and largest opening, respectively. The high resolution
of the smallest crack (i.e. length and points about
$12\text{\,}\mathrm{c}\mathrm{m}$ and 500 points, respectively) enables more
geometrical information available for detection. For the detected cracks,
PtCrack3D achieved a performance of $cr_{con}=0.94$ and $cr_{pre}=0.71$ which
is on par with the Kangaroo Point dataset.

Fig. 6 shows a visualisation of the point cloud output on a surface sample
from the Coolum dataset which represents the detection behaviour in general.
The FPs (in cyan) are similar to the behaviour on Kangaroo Point dataset i.e.
mostly around the edges of cracks and in corners with high changes of surface
normals. Achieving to detect all crack instances, PtCrack3D underestimates the
width of a crack with a large width as highlighted in the yellow box of Fig.
6. This is because the opening width is more than
$35\text{\,}\mathrm{c}\mathrm{m}$ at the widest point, whereas the largest
opening on the dataset used to train PtCrack3D is only about
$10\text{\,}\mathrm{c}\mathrm{m}$.

Fig. 4 shows the point-wise performance curves (in dash-dotted lines) for the
trained PtCrack3D method implementation on the Coolum dataset. A small
absolute drop in recall and specificity (i.e. at threshold of 0.5) of about
14% can be observed compared to the Kangaroo Point test performance. The
precision increased by about 20%. However, this increase in performance
reflects the higher crack to non-crack point ratio of the Coolum dataset.
Normalising the Coolum precision score by equalising its FP rate to the
Kangaroo Point performance, shows an absolute drop of about 18% compared to
the expected precision scores given by the differences in crack points ratio.
The drop in performance across all measured point-wise metrics is expected as
the model are trained using a different dataset. As mentioned earlier, it is
more important to be able to detect cracks as an object, whereas the point-
wise detection enables fine details measurements to be extracted from detected
cracks.

Figure 6: Coolum dataset sample surface point cloud output of PtCrack3D.
Coloured point cloud of a surface with TP (blue), FN (red) and FP (cyan).

##  VI Conclusions

In this paper, we proposed PointCrack3D, a point-cloud-based crack detection
method capable of automatically detecting crack instances from 3D LIDAR data
in highly unstructured surfaces such as rock cliffs or natural/mine tunnels.
This is the first crack detection method utilising a point-based deep learning
architecture for detection on unstructured surfaces. PtCrack3D includes a
method which down-samples a point cloud while maintaining the point density of
cracks and a post-processing step to segment cracks via clustering of 3D
points corresponding to the detected crack instances.

PtCrack3D was first experimentally validated on a dataset of LIDAR point
clouds capturing a large natural rock cliff. The method was shown to detect
97% of the crack instances in the test set overall, and 100% of cracks with a
maximum width of more than $3\text{\,}\mathrm{c}\mathrm{m}$, where the SOTA
MPFH defects detection method could only detect 47% (and 61%, resp.) of those
cracks. PtCrack3D was then applied ‘out of the box’ to a second dataset
acquired in a distinct location and managed to detect all cracks, indicating
its detection capability on an out-of-sample case. The method was evaluated in
different parameters configuration that could be fine-tuned to fit detection
requirements. As an example, setting a high confidence threshold value may be
suitable for applications where small cracks are less important to be
identified whilst focusing on larger cracks and keeping false alarms low. It
could also be concluded that the detection rate depends both on the width of a
crack and the number of points that makes up the crack. Automatically
detecting cracks in highly unstructured surfaces using PtCrack3D is a critical
step towards the deployment of robots for autonomous crack inspection in an
unstructured environment.

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# On groups with few subgroups not in the Chermak–Delgado lattice

David Burrell111David Burrell, University of Florida<EMAIL_ADDRESS>William Cocke222William Cocke, Augusta University School of Computer and Cyber
Sciences<EMAIL_ADDRESS>Ryan McCulloch333Ryan McCulloch, Elmira College,
<EMAIL_ADDRESS>
(Nov 2022)

###### Abstract

We investigate the question of how many subgroups of a finite group are not in
its Chermak–Delgado lattice. The Chermak–Delgado lattice for a finite group is
a self-dual lattice of subgroups with many intriguing properties. Fasolă and
Tărnăuceanu [FT22] asked how many subgroups are not in the Chermak–Delgado
lattice and classified all groups with two or less subgroups not in the
Chermak–Delgado lattice. We extend their work by classifying all groups with
less than five subgroups not in the Chermak–Delgado lattice. In addition, we
show that a group with less than five subgroups not in the Chermak–Delgado
lattice is nilpotent. In this vein we also show that the only non-nilpotent
group with five or fewer subgroups in the Chermak–Delgado lattice is $S_{3}$.

##  1 Introduction.

In this paper we examine the question of how many subgroups of a finite group
$G$ are not in the Chermak–Delgado lattice of $G$. Chermak and Delgado [CD89]
first defined the Chermak–Delgado lattice in 1989. In 2022, Fasolă and
Tărnăuceanu [FT22] investigated the question of how large the Chermak–Delgado
lattice of a finite group can be. They classified all groups with at most $2$
subgroups not in their Chermak–Delgado lattice. In Theorem C, we extend their
work by classifying all groups with less than five subgroups not in their
Chermak–Delgado lattice.

Throughout the paper, we will use the notation $H\leq G$ to mean that $H$ is a
subgroup of $G$. The Chermak–Delgado lattice is a sublattice of the subgroup
lattice of $G$. To define the Chermak–Delgado lattice of a finite group $G$,
which we write as $\operatorname{\mathcal{CD}}(G)$, we first need to define a
function $m_{G}$ called the Chermak–Delgado measure. The function $m_{G}$
takes as input a subgroup of $G$ and returns the product of the order of the
subgroup and the order of its centralizer in $G$, i.e.,

$m_{G}(H)=|H|\cdot|\textbf{C}_{G}(H)|.$

It is surprising that the set of subgroups with maximum Chermak–Delgado
measure have a very special property: they form a lattice, the Chermak–Delgado
lattice.

For a finite group $G$, we write $m^{*}(G)$ for the maximum value of $m_{G}$
on a finite group, i.e.,

$m^{*}(G)=\max_{H\leq G}\left\\{m_{G}(H)\right\\}.$

If the group is clear from context, we will shorten $m^{*}(G)$ to just
$m^{*}$. If $H,K\leq G$ satisfy $m_{G}(H)=m_{G}(K)=m^{*}$, then $HK=\langle
H,K\rangle$ and $m_{G}(HK)=m_{G}(H\cap K)=m^{*}.$ Hence the subgroups with
maximum Chermak–Delgado measure form a sublattice of the subgroup lattice of
$G$: this is the Chermak–Delgado lattice of $G$. The proof that
$\operatorname{\mathcal{CD}}(G)$ is a lattice can be found in Isaacs [Isa08,
1.G].

Before stating our main results, we introduce some new notation. We write
$\delta_{\operatorname{\mathcal{CD}}}(G)$ for the number of subgroups of $G$
not in $\operatorname{\mathcal{CD}}(G)$. Hence,
$\delta_{\operatorname{\mathcal{CD}}}(G)=0$, if all subgroups of $G$ are in
$\operatorname{\mathcal{CD}}(G)$ and
$\delta_{\operatorname{\mathcal{CD}}}(G)=1$ if there is single subgroup of $G$
not in $\operatorname{\mathcal{CD}}(G)$. The cyclic groups show that
$\delta_{\operatorname{\mathcal{CD}}}$ maps onto the natural numbers.

Fasolă and Tărnăuceanu classified all groups where
$\delta_{\operatorname{\mathcal{CD}}}(G)\leq 2$ [FT22, Theorem 1.1] and asked
for such a classification when $\delta_{\operatorname{\mathcal{CD}}}(G)>2$. In
this paper, we provide such a classification when
$\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$. To do this, we first
investigate how $\delta_{CD}(G)$ influences the structure of a finite group
$G$.

######  Theorem A.

Let $G$ be a finite group. If $\delta_{\operatorname{\mathcal{CD}}}(G)<5$,
then $G$ is nilpotent.

Of note, when $\delta_{\operatorname{\mathcal{CD}}}(G)=5$ and $G$ is not
nilpotent, we have the following theorem where $S_{3}$ is the symmetric group
on $3$ symbols.

######  Theorem B.

Let $G$ be a finite group that is not nilpotent. If
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$, then $G\cong S_{3}$.

Using Theorem A together with a mixture of computational and theoretical
results, we are able to complete the classification of groups with
$\delta_{\operatorname{\mathcal{CD}}}(G)<5$.

######  Theorem C.

Let $G$ be a finite group. If $\delta_{\operatorname{\mathcal{CD}}}(G)<5$,
then one of the following holds:

  1. 1.

$\delta_{\operatorname{\mathcal{CD}}}(G)<3$;

  2. 2.

$\delta_{\operatorname{\mathcal{CD}}}(G)=3$ and $G$ is cyclic of order either
$p\cdot q$ or $p^{3}$ for primes $p,q$;

  3. 3.

$\delta_{\operatorname{\mathcal{CD}}}(G)=4$ and $G$ is isomorphic to either a
cyclic group of order $p^{4}$ for a prime $p$, $C_{2}\times C_{2}$, or the
extraspecial group of order 27 and exponent 9.

Hence, we extend the classification of groups with $\delta_{CD}(G)<3$ to
$\delta_{CD}(G)<5$.

The rest of the paper proceeds as follows: In Section 2 we collect some basic
properties of the Chermak–Delgado lattice. In Section 3 we prove Theorem A and
in Section 4 we prove Theorem B. Section 5 contains our proof of Theorem C.
Finally in our conclusion, we ask some questions motivated by the study of
$\delta_{\operatorname{\mathcal{CD}}}$.

##  2 Preliminary material.

In this section we recall a few well-known properties of the Chermak–Delgado
lattice for a finite group $G$. We also present some elementary properties of
the Chermak–Delgado lattice for a few families of groups.

######  Lemma 2.1.

Suppose $G$ is a finite group. If $H\in\operatorname{\mathcal{CD}}(G)$, then
$\textbf{Z}(G)\leq H$.

###### Proof.

If $H\leq G$, then $\textbf{C}_{G}(H)=\textbf{C}_{G}(H\textbf{Z}(G))$, and so
if $H\in\operatorname{\mathcal{CD}}(G)$, and so $H$ attains the maximum
Chermak-Delgado measure in $G$, then $\textbf{Z}(G)\leq H$. ∎

######  Corollary 2.2.

Suppose that a finite nontrivial group $G$ is a $p$-group. Then
$1\notin\operatorname{\mathcal{CD}}(G)$.

###### Proof.

A nontrivial $p$-group has a nontrivial center, and the result follows by
Lemma 2.1. ∎

The following result appears in McCulloch [McC18, Corollary 7].

######  Lemma 2.3.

[McC18] Suppose $G$ is a finite group and
$1\in\operatorname{\mathcal{CD}}(G)$. Then $\operatorname{\mathcal{CD}}(G)$
contains no nontrivial $p$-group for any primes $p$.

We will often use the contrapositive to Lemma 2.3, which we state below as a
lemma.

######  Corollary 2.4.

Suppose $G$ is a finite group. If a nontrivial $p$-group is in
$\operatorname{\mathcal{CD}}(G)$, then
$1\notin\operatorname{\mathcal{CD}}(G)$.

Another well-known result about Chermak–Delgado lattices of finite groups is
the following by Brewster and Wilcox [BW12, Theorem 2.9].

######  Lemma 2.5.

[BW12] Let $G$ and $H$ be finite groups. Then
$\operatorname{\mathcal{CD}}(G\times
H)=\operatorname{\mathcal{CD}}(G)\times\operatorname{\mathcal{CD}}(H)$.

Together with Corollary 2.2 above, this implies that in a finite nilpotent
group we can greatly restrict the types of groups that appear in the
Chermak–Delgado lattice.

######  Corollary 2.6.

Let $G$ be a finite nilpotent group. If $p$ divides $|G|$, then $p$ divides
the order of $H$ for every $H$ in $\operatorname{\mathcal{CD}}(G)$.

###### Proof.

We can write $G$ as $S\times K$ where $S$ is a Sylow $p$-group of $G$ and $K$
is a $p$-complement of $S$. We know that
$\operatorname{\mathcal{CD}}(G)=\operatorname{\mathcal{CD}}(S)\times\operatorname{\mathcal{CD}}(K)$.
Since $\operatorname{\mathcal{CD}}(S)$ does not contain $1$, we conclude that
every element of $\operatorname{\mathcal{CD}}(S)$ is a nontrivial $p$-group.
Hence $p$ divides any $H$ in $\operatorname{\mathcal{CD}}(G)$. ∎

Another interesting result about the Chermak–Delgado lattice is that since
$H\in\operatorname{\mathcal{CD}}(G)$ if and only if
$H^{g}\in\operatorname{\mathcal{CD}}(G)$, we have that $HH^{g}$ is a group for
all $g\in G$. Recall that a subgroup $H$ of a group $G$ is called subnormal if

$H=K_{0}\lhd\,K_{1}\lhd\,\dots\lhd\,K_{n}=G.$

Foguel showed that groups $H$ that permute with their conjugates, i.e.,
$HH^{g}$ is a subgroup, are subnormal [Fog97]. Hence subgroups in
$\operatorname{\mathcal{CD}}(G)$ are subnormal in $G$, which is also noted in
[BW12, Coc20].

The following lemma is part of a theme we will see throughout the paper:
certain conditions on $G$ and $\operatorname{\mathcal{CD}}(G)$ will exclude
other groups from being in $\operatorname{\mathcal{CD}}(G)$. We say that a set
of subgroups $X=\\{H_{1},\dots,H_{k}:H_{i}\leq G\\}$ of cardinality $k$
_witnesses_ that $\delta_{CD}(G)\geq k$, if for each $H_{i}\in X$ we have that
$H_{i}\not\in\operatorname{\mathcal{CD}}(G)$.

######  Lemma 2.7.

Let $G$ be a finite group. If $G$ has $k$ subgroups of prime order, then
$\delta_{CD}(G)\geq k$.

###### Proof.

If $k=0$, then $G$ is trivial and the result is trivially true. So suppose
$k>0$. If $1\in\operatorname{\mathcal{CD}}(G)$, then it follows from Lemma 2.3
that the $k$ subgroups of prime order witness that $\delta_{CD}(G)\geq k$.
Suppose $1\notin\operatorname{\mathcal{CD}}(G)$ and suppose that $H,K\leq G$
are two distinct subgroups of prime order and
$H,K\in\operatorname{\mathcal{CD}}(G)$. Then $1=H\cap
K\in\operatorname{\mathcal{CD}}(G)$, a contradiction. Hence among the
subgroups of prime order, at most one of them can be in
$\operatorname{\mathcal{CD}}(G)$. Thus the other $k-1$ subgroups of prime
order, together with the identity witness that $\delta_{CD}(G)\geq k$. ∎

The following lemma concerns generalized quaternion groups. Recall that a
group is called a generalized quaternion group if $G\cong\langle
a,b|a^{2k},b^{4},a^{k}=b^{2},a^{b}=a^{-1}\rangle.$ We will write the
generalized quaternion group of order $m$ as $Q_{m}$. It is well-known that a
finite $p$-group with a single subgroup of order $p$ is either cyclic or
generalized quaternion [Suz86, 4.4]. We also need the following from Fasolă
and Tărnăuceanu [FT22, The proof of Lemma 2.1].

######  Lemma 2.8 ([FT22]).

If $G$ is a generalized quaternion $2$-group, then

$\left|\operatorname{\mathcal{CD}}\left(Q_{2^{k}}\right)\right|=\begin{cases}5&\text{
if $k=3$}\\\ 1&\text{ if $k>3$.}\end{cases}$

For the next lemma, we will need to introduce some notation. For a group $G$
and two subgroups $H$ and $K$ of $G$, we write $[[H:K]]_{G}$ for the set of
subgroups between $H$ and $K$, i.e.,

$[[H:K]]_{G}=\\{J\leq G:H\leq J\text{ and }J\leq K\\}.$

If the group $G$ is clear from context, then we write $[[H:K]]$ for
$[[H:K]]_{G}$. The following lemma from An [An22a, Theorem 3.4] and [An22b,
Theorem 4.4] shows that when $H\in\operatorname{\mathcal{CD}}(G)$, we can
actually say something abut $\operatorname{\mathcal{CD}}(H)$.

######  Lemma 2.9.

Let $G$ be a finite group and $H\leq G$. If
$H\in\operatorname{\mathcal{CD}}(G)$ then $\operatorname{\mathcal{CD}}(H)$ is
exactly the set of subgroups of $H$ containing $\textbf{Z}(H)$ and in
$\operatorname{\mathcal{CD}}(G)$, i.e.,

$\operatorname{\mathcal{CD}}(H)=[[\textbf{Z}(H):H]]\cap\operatorname{\mathcal{CD}}(G).$

##  3 Proof of Theorem A.

Our proof of Theorem A depends on a connection between the Sylow subgroups and
the Chermak–Delgado lattice. One of the Sylow Theorems states that the total
number of Sylow $p$-subgroups in $G$ is 1 if and only if any of the Sylow
$p$-subgroups is normal in $G$. The below lemma extends this observation.

######  Lemma 3.1.

Let $G$ be a finite group and let $S$ be a Sylow subgroup of $G$. Then $S$ is
subnormal in $G$ if and only if $S$ is normal in $G$.

###### Proof.

A normal subgroup is also subnormal. Suppose now that $S$ is subnormal in $G$,
and that $S$ is a $p$-group for a prime $p$. This means there is a chain of
subgroups

$S=K_{0}\lhd\,K_{1}\lhd\,\dots\lhd\,K_{n}=G.$

We note that $S$ is a Sylow subgroup of all of the $K_{i}$. Since
$S\lhd\,K_{1}$, we know that $S$ is the unique Sylow $p$-subgroup of $K_{1}$.
Hence $S$ is a characteristic subgroup of $K_{1}$. Thus $S$ is normal in
$K_{2}$ and by the same argument characteristic in $K_{2}$. Continuing in this
manner, we see that $S$ is normal in $G$. ∎

######  Lemma 3.2.

Let $G$ be a finite group. Then at most one Sylow subgroup of $G$ is in
$\operatorname{\mathcal{CD}}(G)$. Furthermore, if a Sylow subgroup of $G$ is
in $\operatorname{\mathcal{CD}}(G)$, then it is normal.

###### Proof.

Since all subgroups in $\operatorname{\mathcal{CD}}(G)$ are subnormal, we have
by Lemma 3.1 that if a Sylow subgroup of $G$ is in
$\operatorname{\mathcal{CD}}(G)$, then it is normal.

First suppose that $1\in\operatorname{\mathcal{CD}}(G)$. Now $1$ is a Sylow
subgroup of $G$ for any prime not dividing $|G|$. It follows from Lemma 2.3
that no nontrivial Sylow subgroups of $G$ are in
$\operatorname{\mathcal{CD}}(G)$, and so in this case, $1$ is the unique Sylow
subgroup in $\operatorname{\mathcal{CD}}(G)$.

Suppose $p$ is a divisor of $|G|$ and suppose a Sylow $p$-subgroup $S$ of $G$
is in $\operatorname{\mathcal{CD}}(G)$. Write $|S|=p^{k}$. Since $S$ is normal
in $G$, we have that $S$ is the unique Sylow $p$-subgroup of $G$. Let $T$ be
another Sylow subgroup of $G$, and so $T$ is not a $p$-group. We show that $T$
is not in $\operatorname{\mathcal{CD}}(G)$.

By definition, $S\in\operatorname{\mathcal{CD}}(G)$ means that
$m^{*}(G)=|S|\cdot|\textbf{C}_{G}(S)|$. Since
$1<\textbf{Z}(S)\leq\textbf{C}_{G}(S)$, we conclude that $p^{k+1}|m^{*}(G)$.
Since $S$ is a Sylow $p$-subgroup of $G$ with $|S|=p^{k}$, we know that
$p^{k+1}$ does not divide $|\textbf{C}_{G}(T)|$. And so $p^{k+1}$ does not
divide $|T|\cdot|\textbf{C}_{G}(T)|$ and we conclude that $T$ is not in
$\operatorname{\mathcal{CD}}(G)$. ∎

######  Corollary 3.3.

Let $G$ be a finite group and write $n$ for the number of nontrivial Sylow
subgroups of $G$. Then $\delta_{\operatorname{\mathcal{CD}}}(G)\geq n$.

###### Proof.

If none of the nontrivial Sylow subgroups of $G$ are in
$\operatorname{\mathcal{CD}}(G)$, then the nontrivial Sylow subgroups of $G$
witness that $\delta_{\operatorname{\mathcal{CD}}(G)}\geq n.$

Otherwise, suppose that a nontrivial Sylow subgroup $S$ of $G$ is in
$\operatorname{\mathcal{CD}}(G)$. By Lemma 3.2 and Corollary 2.4,

$\\{T:T\neq S\text{ is a nontrivial Sylow subgroup of $G$}\\}\cup\\{1\\}$

witnesses that $\delta_{\operatorname{\mathcal{CD}}(G)}\geq n.$ ∎

Another fact of Sylow theory is that the number of Sylow $p$-subgroup of a
group $G$ is congruent to $1$ modulo $p$, and this number divides the index in
$G$ of a Sylow $p$-subgroup. Hence if a finite group $G$ is not nilpotent,
then it has at least four Sylow subgroups with equality if and only if $G$ is
a $\\{2,3\\}$-group and the Sylow $3$-subgroup is normal.

###### Prove of Theorem A.

The result for $\delta_{\operatorname{\mathcal{CD}}}(G)<4$ follows from
Corollary 3.3 and the fact that non-nilpotent groups always have at least four
Sylow subgroups. Suppose $\delta_{\operatorname{\mathcal{CD}}}(G)=4$. Let $S$
be the Sylow $3$-subgroup of $G$.

We first argue that $S\in\operatorname{\mathcal{CD}}(G)$. Suppose by way of
contradiction that $S\notin\operatorname{\mathcal{CD}}(G)$. Then the four
subgroups that are not in $\operatorname{\mathcal{CD}}(G)$ are the three Sylow
$2$-subgroups of $G$ and $S$. We conclude that
$1\in\operatorname{\mathcal{CD}}(G)$. It follows from Lemma 2.3 that all of
the nontrivial $2$-groups and $3$-groups of $G$ are not in
$\operatorname{\mathcal{CD}}(G)$. Since
$\delta_{\operatorname{\mathcal{CD}}}(G)=4$, $G$ has exactly four total
nontrivial $2$-groups or $3$-groups, namely the three Sylow $2$-subgroups of
$G$ and $S$. So all nontrivial Sylow subgroups must have prime order. Thus,
$m^{*}(G)=m_{G}(1)=|G|=6$. But $S$ has a Chermak-Delgado measure of $9$, a
contradiction.

Therefore $S\in\operatorname{\mathcal{CD}}(G)$, and the four subgroups that
are not in $\operatorname{\mathcal{CD}}(G)$ are the three Sylow $2$-subgroups
of $G$ and the identity subgroup. We conclude that
$G\in\operatorname{\mathcal{CD}}(G)$.

Note that any Sylow $2$-subgroup of $\textbf{C}_{G}(S)$ is also a Sylow
$2$-subgroup of $G$. This is because $|S|\cdot|\textbf{C}_{G}(S)|=m^{*}(G)$
and since $G\in\operatorname{\mathcal{CD}}(G)$, $|G|$ divides $m^{*}(G)$. Now
we have that $G=ST$ is a direct product where $T$ is a Sylow $2$-subgroup of
$\textbf{C}_{G}(S)$, which implies that $G$ is nilpotent, a contradiction. ∎

We can say more about the non-nilpotent case, as seen in the next section
where we prove Theorem B.

##  4 Proof of Theorem B.

If a finite group $G$ is not nilpotent, then it has at least four Sylow
subgroups, and if $G$ has either four or five Sylow subgroups, then $G$ is a
$\\{2,3\\}$-group. In this section we will sometimes use the non-standard
notation $H_{p}$ to denote a Sylow $p$-subgroup of $G$. This makes our proofs
easier to read by reminding the reader of the prime associated with the Sylow
$p$-subgroup, especially as the prime $p$ will shift between certain lemmata.
In addition, we wish to avoid confusion with $S_{p}$, the symmetric group on
$p$ symbols.

######  Lemma 4.1.

Suppose that a finite $\\{2,3\\}$-group $G$ has exactly four Sylow
$3$-subgroups and a normal Sylow $2$-subgroup $H_{2}$. Then
$\delta_{\operatorname{\mathcal{CD}}}(G)>5$.

###### Proof.

Since $G$ has four Sylow $3$-subgroups, it is not nilpotent. By Theorem A, we
conclude that $\delta_{\operatorname{\mathcal{CD}}}(G)\geq 5$. The four Sylow
$3$-subgroups of $G$ witness that $\delta_{\operatorname{\mathcal{CD}}}(G)\geq
4$. By way of contradiction suppose that
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$.

Note that $4$ divides $|H_{2}|$ which equals the index in $G$ of a Sylow
$3$-subgroup.

Suppose $1\in\operatorname{\mathcal{CD}}(G)$. It follows from Lemma 2.3 that
all of the nontrivial $2$-groups and $3$-groups of $G$ are not in
$\operatorname{\mathcal{CD}}(G)$. Since
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$, $G$ has exactly five total
nontrivial $2$-groups or $3$-groups, namely the four Sylow $3$-groups and the
normal Sylow $2$-group. We conclude that all of the nontrivial Sylow subgroups
have prime order which contradicts $|H_{2}|\geq 4$.

Now suppose that $1\notin\operatorname{\mathcal{CD}}(G)$. By assumption, $1$
and the four Sylow $3$-subgroups are the only subgroups of $G$ not contained
in $\operatorname{\mathcal{CD}}(G)$. Hence $G$ and $H_{2}$ are in
$\operatorname{\mathcal{CD}}(G)$. This means that
$m^{*}(G)=|G|\cdot|\textbf{Z}(G)|$ is divisible by $|G|$. However,
$m^{*}(G)=|H_{2}|\cdot|\textbf{C}_{G}(H_{2})|$. Since $H_{2}$ is a $2$-group,
we conclude that $[H_{2},H_{3}]=1$ for a Sylow $3$-subgroup $H_{3}$ of $G$.
Then $H_{2}H_{3}=G$ is a direct product and is thus nilpotent, a
contradiction. Hence $\delta_{\operatorname{\mathcal{CD}}}(G)>5.$ ∎

######  Lemma 4.2.

Suppose that a finite $\\{2,3\\}$-group $G$ has exactly three Sylow
$2$-subgroups and a normal Sylow $3$-subgroup $H_{3}$. If
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$, then
$1\notin\operatorname{\mathcal{CD}}(G)$.

###### Proof.

Suppose by way of contradiction that $1\in\operatorname{\mathcal{CD}}(G)$. It
follows from Lemma 2.3 that all of the nontrivial $2$-groups and $3$-groups of
$G$ are not in $\operatorname{\mathcal{CD}}(G)$. Since
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$, $G$ has at most five nontrivial
$2$-groups or $3$-groups, including the three Sylow $2$-subgroups and $H_{3}$.
And so $G$ has at most two subgroups that are nontrivial $3$-groups, and at
most one subgroup that is a nontrivial $2$-group, but not Sylow. This means
that $G$ is a $\\{2,3\\}$-group with $|G|\leq 2^{2}3^{2}$. Considering each of
the possible orders of $G$, $2\cdot 3$, $4\cdot 3$, $2\cdot 9$, or $4\cdot 9$,
we see that in every case, the Sylow subgroups are abelian and one of them has
Chermak–Delgado measure larger than $m^{*}(G)=|G|$. ∎

######  Lemma 4.3.

Suppose that a finite $\\{2,3\\}$-group $G$ has exactly three Sylow
$2$-subgroups and a normal Sylow $3$-subgroup $H_{3}$. If
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$, then
$H_{3}\in\operatorname{\mathcal{CD}}(G)$.

###### Proof.

By Lemma 4.2, we know that $1\notin\operatorname{\mathcal{CD}}(G)$. Hence $1$,
together with the three Sylow $2$-subgroups witness that
$\delta_{\operatorname{\mathcal{CD}}}(G)\geq 4$. There is one other subgroup
of $G$ not in $\operatorname{\mathcal{CD}}(G)$.

Suppose by way of contradiction that
$H_{3}\notin\operatorname{\mathcal{CD}}(G)$. Let $H_{2}$ be a Sylow
$2$-subgroup. We claim that one of $H_{3}$ or $H_{2}$ must have prime order.
Otherwise, if $1<P<H_{3}$, and $1<Q<H_{2}$, then
$P,Q\in\operatorname{\mathcal{CD}}(G)$, and so $1=P\cap
Q\in\operatorname{\mathcal{CD}}(G)$, a contradiction. Let $K$ be a Sylow
subgroup of prime order $p$.

Then $K\cap\textbf{Z}(G)=1$, as otherwise, $K\leq\textbf{Z}(G)$, and it would
follow that the $\\{2,3\\}$-group $G$ is nilpotent. And so $\textbf{Z}(G)$ is
a $p^{\prime}$-group. Now
$m^{*}(G)=|G|\cdot|\textbf{Z}(G)|=p\cdot(p^{\prime})^{i}$ for some $i$. But
also $K<K\textbf{Z}(G)<G$, and so
$K\textbf{Z}(G)\in\operatorname{\mathcal{CD}}(G)$. But $K\textbf{Z}(G)$ is
abelian, and so $K\textbf{Z}(G)\leq\textbf{C}_{G}(K\textbf{Z}(G))$, and so
$p^{2}$ divides
$|K\textbf{Z}(G)|\cdot|\textbf{C}_{G}(K\textbf{Z}(G))|=m^{*}(G)$, a
contradiction.

∎

We can now prove Theorem B, which states that for a non-nilpotent group if
$\delta_{\operatorname{\mathcal{CD}}}(G)=5,$ then $G\cong S_{3}$.

###### Proof of Theorem B.

From Lemma 4.1 we conclude that $G$ is a $\\{2,3\\}$ group with exactly three
Sylow $2$-subgroups, and a normal Sylow $3$-subgroup $H_{3}$. From Lemmata 4.2
and 4.3 we have that $H_{3}\in\operatorname{\mathcal{CD}}(G)$ and that the
five subgroups of $G$ not in $\operatorname{\mathcal{CD}}(G)$ are the three
Sylow $2$-subgroups, the identity subgroup, and one other subgroup $X$. We
will show that $X=G$. Suppose instead that
$G\in\operatorname{\mathcal{CD}}(G)$.

Note that any Sylow $2$-subgroup of $\textbf{C}_{G}(H_{3})$ is also a Sylow
$2$-subgroup of $G$. This is because
$|H_{3}|\cdot|\textbf{C}_{G}(H_{3})|=m^{*}(G)$ and since
$G\in\operatorname{\mathcal{CD}}(G)$, $|G|$ divides $m^{*}(G)$. And so now we
have that $G=H_{3}H_{2}$ is a direct product where $H_{2}$ is a Sylow
$2$-subgroup of $\textbf{C}_{G}(H_{3})$, which implies that $G$ is nilpotent,
a contradiction.

So $X=G$, and the three Sylow $2$-subgroups, the identity subgroup, and $G$
witness that $\delta_{\operatorname{\mathcal{CD}}}(G)=5$.

Let $H_{2}$ be a Sylow $2$-subgroup of $G$. We argue that $|H_{2}|=2$.
Otherwise, there exists $1<X\leq\textbf{Z}(H_{2})$ with $X<H_{2}$, and so
$X\in\operatorname{\mathcal{CD}}(G)$. Then
$m^{*}(G)=|X|\cdot|\textbf{C}_{G}(X)|=|H_{3}|\cdot|\textbf{C}_{G}(H_{3})|$,
and so $|H_{3}|$ divides $|\textbf{C}_{G}(X)|$. This implies that
$H_{3}\leq\textbf{C}_{G}(X)$. Hence $[X,H_{2}]=1$ and $[X,H_{3}]=1$. Since
$\textbf{Z}(G)\leq H_{3}$, we have $X\leq\textbf{Z}(G)\leq H_{3}$, a
contradiction. Hence $|H_{2}|=2$. Since $|H_{2}|=2$, we note that
$\textbf{C}_{G}(H_{3})$ is a $3$-group as otherwise a Sylow $2$-subgroup would
be contained in $\textbf{C}_{G}(H_{3})$ which would imply that $G$ is
nilpotent. So $m^{*}(G)=|H_{3}|\cdot|\textbf{C}_{G}(H_{3})|$ is a power of
$3$.

We now argue that $H_{2}$ is self-normalizing. Otherwise, let $Y$ be a
nontrivial Sylow $3$-subgroup of $\textbf{N}_{G}(H_{2})$. So $1<Y<H_{3}$ (note
that $Y<H_{3}$ as otherwise $H_{3}$ and $H_{2}$ would normalize one another,
and so $G=H_{3}H_{2}$ would be nilpotent). Then $H_{2}Y$ is a subgroup, and
$H_{2}<H_{2}Y<G$, and so $H_{2}Y$ is in $\operatorname{\mathcal{CD}}(G)$,
which contradicts the fact that $m^{*}(G)$ is a power of $3$.

So $H_{2}$ is self-normalizing, and
$3=|G:\textbf{N}_{G}(H_{2})|=|G:H_{2}|=|H_{3}|$. So $H_{3}$ is cyclic of order
$3$, $H_{2}$ is cyclic of order $2$, and $G\cong S_{3}$. ∎

Figure 1 shows the subgroup diagram of $S_{3}$.

$S_{3}$$C_{2}$$C_{2}$$C_{2}$$C_{3}$$1$ Figure 1: Subgroup diagram of $S_{3}$
with a circle around the Chermak–Delgado sublattice, which consists of the
single subgroup isomorphic to $C_{3}$. This is the only non-nilpotent group
with $\delta_{\operatorname{\mathcal{CD}}}(G)=5$.

##  5 Proof of Theorem C.

######  Lemma 5.1.

Let $G$ be a nonabelian nilpotent group. If
$\delta_{\operatorname{\mathcal{CD}}}(G)<7,$ then $G$ is a $p$-group. If
$\delta_{\operatorname{\mathcal{CD}}}{G}<6$, then $G$ is a $2$-group or a
$3$-group.

###### Proof.

Suppose that $G$ is not a $p$-group, i.e., there are at least two prime
divisors of $G$. Corollary 2.6 tells us that no $p$-groups are in
$\operatorname{\mathcal{CD}}(G)$. Moreover, since $G$ is nonabelian, then at
least one Sylow subgroup $S$ of $G$ is nonabelian. Nonabelian $p$-groups have
at least $p+2$ subgroups above the center, which means they have at least
$p+4$ subgroups total. Thus $G$ contains $p+4$ distinct $p$-groups for the
prime $p$.

Together with a nontrivial Sylow $q$-group $T$, these form a set of $p+5$
subgroups that witness $\delta_{\operatorname{\mathcal{CD}}}(G)\geq p+5$.

Thus we have

$p+5\leq\delta_{\operatorname{\mathcal{CD}}}(G)<7,$

and we conclude that $G$ must be a $p$-group.

Recall that a nonabelian $p$-group of odd order must contain $p+1$ subgroups
of order $p$. By Lemma 2.7 this means that

$p+1\leq\delta_{\operatorname{\mathcal{CD}}}(G)<6$

and we conclude that $p<4.$

∎

We note that for an odd prime $p$,
$\delta_{\operatorname{\mathcal{CD}}}(Q_{8}\times C_{p})=7$ and is a
nonabelian nilpotent group. We also note that the extraspecial group of order
$5^{3}$ and exponent $25$ satisfies
$\delta_{\operatorname{\mathcal{CD}}}(G)=6$, so both bounds in Lemma 5.1 are
sharp.

######  Lemma 5.2.

Let $G$ be a nonabelian $p$-group and suppose that
$3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$. Then $G$ contains at
least $p+1$ subgroups of order $p$. Of these, $p$ of the subgroups together
with the identity witness that $\delta_{CD}(G)\geq p+1$.

###### Proof.

If $p>2$, then every nonabelian $p$-group contains at least $p+1$ subgroups of
order $p$.

If $p=2$, and $G$ contains a single subgroup of order $2$, then $G$ is
generalized quaternion. Since the generalized quaternion 2-groups of order
greater than $8$ contain more than 7 subgroups and contain a single subgroup
in their Chermak–Delgado lattices, we conclude that for such groups
$\delta_{\operatorname{\mathcal{CD}}}(G)>4$. Moreover
$\delta_{\operatorname{\mathcal{CD}}}(Q_{8})=1$. ∎

Hence for a nonabelian $p$-group $G$ with
$3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$ there are $p$ subgroups
of order $p$, which together with the identity witness that
$\delta_{CD}(G)\geq p+1$. We will use this to derive a set of conditions for a
subgroup of $G$ to satisfy.

######  Lemma 5.3.

Let $G$ be a nonabelian $p$-group and suppose
$3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$. For every $K\leq G$ the
following hold:

  1. 1)

There are at most $4$ subgroups of order $p$ in $K$.

  2. 2)

There is a subgroup $Z\leq\mathbf{Z}(K)$ with $|Z|=p$ such that

$|\\{H\leq K:p^{2}\leq|H|\text{ and }Z\not<H\\}|\leq\begin{cases}0&p>2\\\
1&p=2.\end{cases}$

  3. 3)

If $|K|\geq p^{3}$, then $K\in\operatorname{\mathcal{CD}}(K)$ and at most one
subgroup $H$ of $K$ can satisfy $\textbf{Z}(K)\leq H$ and
$H\notin\operatorname{\mathcal{CD}}(K)$.

###### Proof.

As noted in Lemma 5.2, the group $G$ has at least $p+1$ subgroups not in
$\operatorname{\mathcal{CD}}(G)$ consisting of the identity and the $p$
subgroups of order $p$. 1) If $K$ had more than 4 subgroups of order $p$, then
it would have at least $5$ such groups. Hence $G$ would have at least 5
subgroups of order $p$ and by Lemma 2.7 we would have that
$\delta_{\operatorname{\mathcal{CD}}}(G)\geq 5$.

2) If $K$ has a single subgroup of order $p$ it is cyclic or generalized
quaternion and all groups of order greater than $p^{2}$ contain the unique
subgroup of order $p$. Now suppose that $K$ has $p+1$ involutions. Hence $G$
has $p+1$ witnesses from Lemma 5.2. There can only be one additional witness
with order greater than $p$ if $p=2$.

Suppose that $p=2$. Then $K$ contains the three subgroups $A=\langle
a\rangle$, $B=\langle b\rangle$, and $C=\langle ab\rangle$ of order 2. If none
of them are in $\operatorname{\mathcal{CD}}(G)$, then the four subgroups not
in $\operatorname{\mathcal{CD}}(G)$ are $\\{1,A,B,C\\}$. Now $\langle
a,b\rangle$ contains a central involution, and hence is a product of any two
of the subgroups of order $2$ and has order $4$. Now $\langle a,b\rangle$ is
in $\operatorname{\mathcal{CD}}(G)$, and it follows that
$\textbf{Z}(G)=\langle a,b\rangle$. Moreover, since
$\delta_{\operatorname{\mathcal{CD}}}(G)=4$ in this case, we conclude that
every subgroup of $G$ of order $4$ or larger is in
$\operatorname{\mathcal{CD}}(G)$, and thus contains $\textbf{Z}(G)=\langle
a,b\rangle$.

Otherwise let $Z\leq K$ of order 2 in $\operatorname{\mathcal{CD}}(G)$. Since
$Z\in\operatorname{\mathcal{CD}}(G)$, we know that $Z=\mathbf{Z}(G)$ and hence
$Z\leq\mathbf{Z}(K)$. Moreover, since the other two involutions and the
identity would not be in $\operatorname{\mathcal{CD}}(G)$, we have
$\delta_{\operatorname{\mathcal{CD}}}(G)\geq 3$ and thus at most one other
subgroup of $G$ (which must have order greater than 2) could not be in
$\operatorname{\mathcal{CD}}(G)$ and thus not contain $Z$.

Suppose that $p>2$. Then by Lemma 5.1, $p=3$ and from 5.2 we note that exactly
one subgroup of order $3$ is in $\operatorname{\mathcal{CD}}(G)$; call this
subgroup $H$ and note that $H=\textbf{Z}(G)$. Then every subgroup of $G$ of
order greater than $3$ is in $\operatorname{\mathcal{CD}}(G)$ and must contain
$H$ as a subgroup.

3) We note that if $K$ is cyclic then
$\mathbf{Z}(K)=K\in\operatorname{\mathcal{CD}}(K)$. Suppose that $|K|\geq
p^{3}$ is not cyclic and by way of contradiction that
$K\not\in\operatorname{\mathcal{CD}}(K)$. By Lemma 2.9 we have that
$K\not\in\operatorname{\mathcal{CD}}(G)$. Moreover, since
$\operatorname{\mathcal{CD}}(G)$ is closed under products we must have that at
least $p$ of the maximal subgroups of $K$ (which have order $\geq p^{2}$) are
not in $\operatorname{\mathcal{CD}}(G)$. Hence
$\delta_{\operatorname{\mathcal{CD}}}(G)\geq 2(p+1)>5$. We conclude that
$K\in\operatorname{\mathcal{CD}}(K)$.

Suppose that $H$ and $J$ are subgroups of $K$ with $\mathbf{Z}(K)\leq H$ and
$\mathbf{Z}(K)\leq J$ such that neither $H$ nor $J$ are in
$\operatorname{\mathcal{CD}}(K)$. Then $H$ and $J$ are not in
$\operatorname{\mathcal{CD}}(G)$ by Lemma 2.9. Since
$K\in\operatorname{\mathcal{CD}}(K)$, we conclude that $\mathbf{Z}(K)<H$ and
$\mathbf{Z}(K)<J$ and thus the orders of both $H$ and $J$ are greater than or
equal to $p^{2}$. Hence, $H$, $J$, the identity, and $p$ of the subgroups of
order $p$ in $G$ would witness that
$\delta_{\operatorname{\mathcal{CD}}}(G)\geq p+3\geq 5$. ∎

Lemma 5.3 gives us conditions on the types of subgroups a $2$-group or
$3$-group $G$ can have if $3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq
4$. This allows us to specify what types of subgroups $G$ can have of a given
order. For example, a $2$-group $G$ with
$3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$ cannot contain the
dihedral group of order $8$ as a subgroup because the dihedral group of order
$8$ has 5 involutions. Similarly, it cannot contain the elementary abelian
group of order $8$ as a subgroup. This means all of its subgroups of order $8$
are either cyclic, quaternion, or isomorphic to $C_{4}\times C_{2}$. We can
computationally then search over all of the $2$-groups of a given order to
look for potential subgroups of a 2-group $G$ with
$3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4.$

We will pause to introduce the reader to another family of $p$-groups. These
groups are commonly written as $M_{p^{k}}=\langle
a,b|a^{p^{k-1}},b^{p},a^{b}=a^{p^{k-2}+1}\rangle.$ For an odd prime $p$, the
groups $M_{p^{k}}$ are the only nonabelian $p$-groups with cyclic maximal
subgroups.

######  Lemma 5.4.

Let $G$ be a $2$-group and suppose
$3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$. Then all subgroups of
order $32$ in $G$ are isomorphic to one of the following groups:
$C_{32},\,C_{16}\times C_{2},\,$ or $M_{32}$.

###### Proof.

Using GAP [GAP22] we can sort through all groups of a given order that meet
the conditions of Lemma 5.3. For groups of order 32, we obtain the Venn
diagram in Figure 2. Code to do this in the GITHUB repo:
https://github.com/7cocke/chermak_delgado_lt5. Different authors wrote the
code independently of each other in the two languages.

Figure 2: Groups of order 32 that satisfy at least one of conditions 1), 2),
and 3) of Lemma 5.3 identified by their SmallGroup index among groups of order
32, e.g., $1$ corresponds to SmallGroup(32,1) which is the cyclic group of
order 32.

∎

We could continue our calculations and would see that there are 3 possible
subgroups of order 64 which are $C_{64},C_{32}\times C_{2}$ and $M_{64}$.
Similarly, we would see that there are 3 possible subgroups of order 128,
i.e., the groups $C_{128},C_{64}\times C_{2},M_{128}.$ There are also 3
possible subgroups of order 256. This observation motivates the following
theorem whose proof relies heavily on the exhaustive work of Berkovich and
Janko [BJ06].

######  Theorem 5.5.

Let $G$ be a $p$-group with order $|G|=p^{k+1}$ for $k>4$. Suppose that $G$
contains at most p+1 subgroups of order $p$ and that every maximal subgroup of
$G$ is isomorphic to one of $C_{p^{k}},C_{p^{k-1}}\times C_{p}$, and
$M_{p^{k}}$. Then $G$ is isomorphic to one of the groups
$C_{p^{k}+1},C_{p^{k}}\times C_{p}$, or $M_{p^{k+1}}$.

###### Proof.

If $G$ is abelian, then it must be isomorphic to either $C_{p^{k}}$ or
$C_{p^{k-1}}\times C_{p}$.

The rest of the proof will cite a number of results from the encyclopedic work
of Berkovich and Janko [BJ06] which classifies $p$-groups all of whose
subgroups of index $p^{2}$ are abelian.

Suppose that all maximal subgroups of $G$ are abelian. Then $G$ is minimal
nonabelian. Berkovich and Janko [BJ06, 3.1] attribute the classification of
minimal nonabelian $p$-groups to Rédei. Our group $G$ contains at most $p+1$
subgroups of order $p$ and has order greater than $p^{4}$. If $G$ were minimal
nonabelian, it would have to be isomorphic to $M_{p^{k+1}}$.

Hence $G$ must contain a nonabelian maximal subgroup. Thus $G$ is an
$A_{2}$-group, i.e., a group all of whose subgroups of index $p^{2}$ are
abelian. Moreover, all proper subgroups of $G$ would be metacyclic. If $G$
itself were not metacyclic, then it would be minimal nonmetacyclic. Berkovich
and Janko [BJ06, 1.1(l)] cite Blackburn [Bla61, 3.2] for the classification of
minimal nonmetacyclic groups, none of which could be our group $G$. Hence the
group $G$ is a metacyclic $A_{2}$-group. Such groups are classified by
Berkovich and Janko [BJ06, 5.2]. Again, since we have restricted the maximal
subgroups of $G$, it cannot be any of these groups.

We conclude that $G$ is either abelian or isomorphic to $M_{p^{k+1}}$. ∎

For $3$-groups, Lemma 5.3 can be used to establish the following.

######  Lemma 5.6.

Let $G$ be a nonabelian $3$-group with
$\delta_{\operatorname{\mathcal{CD}}}(G)=4$. Then all subgroups of order $243$
in $G$ are isomorphic to one of $C_{243},\,C_{81}\times C_{3},\,$ or
$M_{243}$.

These are the same 3 types of groups that occur in Theorem 5.5. This means
that as in the 2-group case, for a $3$-group with
$\delta_{\operatorname{\mathcal{CD}}}(G)=4,$ there are at most three
isomorphism classes of subgroups for every order greater than 243.

######  Theorem 5.7.

If $G$ is a nonabelian $2$-group and
$\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$, then $G$ is the quaternion
group of order $8$.

###### Proof.

If $G$ has a single involution, then $G$ is generalized quaternion and we note
that $m^{*}(G)=\left(\frac{|G|}{2}\right)^{2}$. If $|G|>8$, then
$\operatorname{\mathcal{CD}}(G)$ is a single subgroup and
$\delta_{\operatorname{\mathcal{CD}}}(G)>4.$

Suppose by way of contradiction that $G$ is not the quaternion group of order
$8$. We must have that $3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$.
As noted, in Lemma 5.4 the only subgroups of order $32$ in $G$ would be
isomorphic to $C_{32},C_{16}\times C_{2}$, or $M_{32}$. By Theorem 5.5, the
only subgroups of order 64 in $G$ would be isomorphic to $C_{64},C_{32}\times
C_{2}$ or $M_{64}$. Continuing in this manner, we see that $G$ itself would be
isomorphic to either $C_{2^{k}},C_{2^{k-1}}\times C_{2}$ or $M_{2^{k}}$.
However, none of these groups satisfy
$\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$.

Thus $|G|\leq 32$. A computational search using either GAP or MAGMA shows that
there is no such nonabelian $2$-group. ∎

We note that the dihedral group of order $8$ has exactly $5$ groups not
contained in $\operatorname{\mathcal{CD}}(G)$, i.e.,
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$.

######  Theorem 5.8.

If $G$ is a nonabelian $3$-group, then
$\delta_{\operatorname{\mathcal{CD}}}(G)\geq 4$ with equality if and only if
$G$ is the extraspecial group of order 27 and exponent $9$.

###### Proof.

Using the same code as in Lemma 5.4 we see that all of the subgroups of $G$ of
order 243 are isomorphic to $C_{243},C_{81}\times C_{3}$, or $M_{243}$. By
Theorem 5.5, either $|G|\leq 81$ or $G$ itself is isomorphic to one of
$C_{3^{k}},C_{3^{k-1}}\times C_{3},$ or $M_{3^{k}}$. Regardless, $G$ does not
satisfy $\delta_{\operatorname{\mathcal{CD}}}(G)=4$ except for when $G$ is
$M_{27}$ which is the extraspecial group of order 27 and exponent 9. ∎

In Figure 3 we display the subgroup diagram of $M_{27}$ the only nonabelian
nilpotent group with $\delta_{CD}(G)=4$.

$M_{27}$$C_{9}$$C_{9}$$C_{9}$$C_{3}\times
C_{3}$$C_{3}$$C_{3}$$C_{3}$$C_{3}$$1$ Figure 3: The subgroup diagram of
$G=M_{27}$, i.e., the extraspecial group of order 27 and exponent 9. The four
subgroups not in $\operatorname{\mathcal{CD}}(G)$ are contained wthin the
ellipse in the figure. This is the only nonabelian group with
$\delta_{CD}(G)=4$.

Combining Theorems 5.7 and 5.8 provides the proof of Theorem C.

###### Proof of Theorem C.

If $G$ is nonabelian, then $G$ is either a $2$-group or a $3$-group by Lemma
5.1. Combining Theorem 5.7 and 5.8, we see that $G$ must be the extraspeical
group of order $27$ and exponent $9$.

If $G$ is abelian, then we know that $\delta_{\operatorname{\mathcal{CD}}}(G)$
is equal to the number of subgroups of $G$ minus 1. Counting the subgroups of
abelian groups finishes the proof. ∎

##  6 Conclusion.

We have extended the results by Fasolă and Tărnăuceanu [FT22] by classifying
all finite groups with $3\leq\delta_{\operatorname{\mathcal{CD}}}(G)\leq 4$.
Obvious questions exist about classifying all groups where
$\delta_{\operatorname{\mathcal{CD}}}(G)=k$ for $k\geq 5$. In particular, for
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$ combining Theorem B and Lemma 5.1
means one only needs to classify nonabelian $2$-groups and $3$-groups with
$\delta_{\operatorname{\mathcal{CD}}}(G)=5$.

Corollary 3.3 establishes that $\delta_{\operatorname{\mathcal{CD}}}(G)$ is
greater than the number of distinct prime divisors of a group $G$. Is it the
case that for a fixed $k$ we can also bound the multiplicity of a prime
divisor of $G$? We ask this in three distinct, but highly related questions:

######  Question 1.

Let $n$ be a positive integer not equal to 8. If $G$ has order $n$, then is
$\delta_{\operatorname{\mathcal{CD}}}(G)\geq\delta_{\operatorname{\mathcal{CD}}}(C_{n})$
= number of proper divisors of $n$.

######  Question 2.

Let $G$ be a finite group. If $G$ has exactly $k$ prime divisors up to
multiplicity, is it the case that $\delta_{\operatorname{\mathcal{CD}}}(G)$ is
bounded by a function depending on $k$?

######  Question 3.

Let $G$ be a finite group. If $H\leq G$ has order $p^{k}$ for some prime $p$
and integer $k$, then is

$\delta_{\operatorname{\mathcal{CD}}}(G)\geq\begin{cases}k&\text{ $p$ is
odd,}\\\ k-2&\text{ $p=2$.}\end{cases}$

Theorem A shows that when $\delta_{\operatorname{\mathcal{CD}}}(G)<5$, we have
that $G$ is nilpotent. One can ask the same question for solvable groups.

######  Question 4.

What is the maximal value of $m$, such that if a finite group satisfies
$\delta_{\operatorname{\mathcal{CD}}}(G)<m$, then $G$ must be solvable?

We note that Theorem C shows that there is no nonabelian group $G$ with
$\delta_{CD}(G)=3$.

######  Question 5.

Is there a positive integer $m$ such that for every $k\geq m$ there is a
nonabelian group $G$ with $\delta_{\operatorname{\mathcal{CD}}}(G)=m$.

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