# Theory of the magnetic and metal-insulator transitions in RNiO3 bulk and
layered structures.

Bayo Lau and Andrew J. Millis  Department of Physics, Columbia University, 538
West 120th Street, New York, NY, USA 10027

###### Abstract

A slave rotor-Hartree Fock formalism is presented for studying the properties
of the p-d model describing perovskite transition metal oxides, and a flexible
and efficient numerical formalism is developed for its solution. The
methodology is shown to yield, within an unified formulation, the significant
aspects of the rare earth nickelate phase diagram, including the paramagnetic
metal state observed for the LaNiO3 and the correct ground-state magnetic
order of insulating compounds. It is then used to elucidate ground state
changes occurring as morphology is varied from bulk to strained and un-
strained thin-film form. For ultrathin films, epitaxial strain and charge-
transfer to the apical out-of-plane oxygen sites are shown to have significant
impact on the phase diagram.

###### pacs:

71.30.+h,73.21.-b,75.25.-j,75.25.Dk

Understanding the unusual electronic behavior of transition-metal oxides has
been a long-standing question in condensed matter physics,mit  and interest
has intensified following the demonstration Ohtomo02  that the materials could
be used as components of atomically precise oxide heterostructures. Mannhart08
; Hwang12 ; Chakhalian12  The theoretical challenge posed by the materials is
to treat simultaneously the strong local correlations in the transition metal
d-orbitals and their substantial hybridization with oxygen $p$ orbitals. In
some systems, the $p$ orbitals can be integrated out and the physics
represented in terms of a (possibly multi band) Hubbard model representing the
$d$ orbitals only, for which many theoretical methods are availablemit .
However, in many cases the charge transfer between $p$ and $d$ orbitals is
large enough that the $p$ states cannot be neglected. This “negative charge
transfer energy” regimezsa  has been less extensively studied. While much
useful information has been provided by density functional theory (DFT)
Jones89  and extensions such as DFT+UAnisimov93 ; dft0 ; dft1 ; Han11 ; Han12
; Blanca-romero11 ; Anisimov99 , hybrid functionals Gou11 ; Puggioni12 , and
DFT+DMFT Kotliar06 ; Park12 , these methods are computationally intensive, so
that the large supercells required for long-period ordered phases are
difficult to study. Furthermore, the variety of experimental bulk and
superlattice configurations and of many-body phenomena emphasizes the need for
a model-system treatment that encapsulates the essential physics so the
importance of different contributions can be disentangled.

Figure 1:  (Top to bottom) Bulk, 2-, and 1-layer phase diagram on the
$u$-$\delta d_{pd}$ plane. Slight charge ordering accompanies the spin
ordering for $\delta d_{pd}\neq$0\. $\Uparrow$/$\uparrow$/0 denote large
moment/small moment/non-magnetic sites of the long-period magnetic ordering
patterns detailed in the text. In the lower two panels the light dotted line
indicates the phase boundary obtained without the energy shift on the apical
oxygen.

The rare earth nickelatesnio3_rev , $ReNiO_{3}$, are an important case in
point. According to standard valence counting arguments, Ni is in the 3+
oxidation state with a $d^{7}$ valence configuration. However, photoemission
experiments mizokawa  reported that the dominant GS configuration is
$d^{8}\underline{L}$, in agreement with unrestricted Hartree-Fock
(UHF)mizokawa ; hf1  and DFT+UBlanca-romero11 ; Han12 ; Anisimov99 , and
DFT+DMFTPark12  approaches. Additionally, O 1s x-ray absorption experiments
have found significant hole concentration on the oxygen $p$ orbitalsmizokawa ;
xas0 ; xas1 . This large degree of charge transfer places the material near
the negative charge-transfer regimezsa ; hf1 . As $Re$ is varied across the
lanathanum row of the periodic table, the ground state (GS) of bulk materials
changes from paramagnetic metal (PM-M) to correlated insulator mit ; nio3_rev
; Torrance92 . The correlated insulator phases exhibit a rock-salt-pattern
lattice distortion garcia1 ; alonso0 ; alonso1 ; medarde0 ; Medarde08  and a
nontrivial long-period magnetic ordering alonso0 ; alonso1 ; garcia0 ;
scagnoli0 ; scagnoli1 ; bodenthin . In ultra-thin films a metal-insulator
transition which is apparently unaccompanied by rock-salt-pattern lattice
distortion occurs as film thickness and strain are varied liu ; liu1 ; keimer
; chakhalian ; son0 ; son1 ; moon . Understanding how these apparently
different transitions can occur within a single formulation is an important
open theoretical challengedft0 ; dft1 ; mazin ; Gou11 ; Han11 ; Han12 ;
Puggioni12 ; Blanca-romero11 ; chakhalian ; Anisimov99 ; hf1 ; balents_prl ;
balents_prb ; Park12 .

In this paper, we develop a mean-field (MF) approach based on a combination of
slave-rotor (SR)rotor0 ; rotor1 ; rotor2 ; rotorj0 ; rotorj1  and Hartree-Fock
(HF) methods, along with an efficient and flexible numerical formalism for
solving the MF equations. The method is powerful enough to permit the
examination of the large supercells needed to investigate long-period ordering
patterns in the context of realistic crystal structures. The results reconcile
the bulk and film phase diagrams and allows us to identify the key role played
by the oxygen degrees of freedom and the lattice distortions.

Figure 1 summarizes our key new results. It presents phase diagrams in the
space of Ni charging energy $U$ and amplitude $\delta d_{pd}$ of rocksalt-type
lattice distortion (defined more precisely below). The upper panel shows that,
in the absence of oxygen breathing distortion, bulk materials are PM-M at any
value of $U$, consistent with experimentnio3_rev . As the lattice is
distorted, transitions occur, first to a metallic magnetic phase and then to a
magnetic insulator. The nontrivial ordering wavevector found in
experimentalonso0 ; alonso1 ; garcia0 ; scagnoli0 ; scagnoli1 ; bodenthin  is
correctly obtained as a 3D 16-formula-unit magnetic pattern of the type
$\Uparrow$0$\Downarrow$0, denoting the z-projection of magnetic moment
detailed below. The 4 $\Uparrow$ and 4 $\Downarrow$ Ni sites are surrounded by
6 spin-0 Ni sites, which in our interpretation form singlets with its 6 oxygen
neighbors. Each of the 8 spin-0 Ni sites are connected to 3 $\Uparrow$ and 3
$\Downarrow$ Ni sites via oxygen bonds. States with the
$\uparrow\uparrow\downarrow\downarrow$ pattern are unstable against PM or
$\Uparrow 0\Downarrow 0$ solutions, and indeed are not observed in
experimentsscagnoli0 . In agreement with symmetry-based argumentsbalents_prl ,
slight charge ordering always accompanies the spin ordering. For reasonable
parameter values the change in electronic structure across the $Re$ series is
properly accounted for, with one exception: the magnetic metallic phase found
in theory is not observed in experiment. This is discussed in more detail
below. The lower panels show the evolution of the phase diagram with film
thickness $M$ and (bottom panel) with applied strain, revealing that in ultra-
thin films an insulating phase can occur even in the absence of rocksalt-type
lattice distortions.

Our theoretical approach begins from a $p$-$d$ lattice model of the form
$H=\sum_{k}H^{pd}_{k}+\sum_{l}\left(U_{l}+J_{l}\right)$, with $k$ and $l$ sum
over crystal momenta and Ni sites, respectively. The parameters of this model
may be obtained from, e.g. maximially localized Wannier fits to DFT
resultsMarzari97 , but the precise form is not important for this paper (see
also Ref. Wang11,  for another example of the insensitivity of results to the
precise p-d model band parameters). We adopt a simplified scenario in which we
retain the Ni $e_{g}$ and O $2p\sigma$ orbitals, with nearest-neighbor $p$-$d$
and $p$-$p$ hopping. The difference between the bare Ni-d and O-2p energies
$\varepsilon_{d}$-$\varepsilon_{p}$ is important, as discussed below.

The structure of the ReNiO3 materials is derived from the ideal ABO3 cubic
perovskite structure which is a lattice of corner-sharing oxygen octahedra,
each containing a Ni site at the center. The crystal structure of the actual
materials is distorted from this structure by rotations of the octahedra which
are not important for our purposes and, in the insulating cases, by a two-
sublattice distortion in which adjacent Ni’s have significant different Ni-O
bond lengthsgarcia1 ; alonso0 ; alonso1 ; Medarde08 . To incorporate the bond
disproportionation in the Hamiltonian under the cubic approximation, we scale
the hopping according to the Harrison ruleharrison ,
$t_{pd}=t_{pd}^{0}\left(1+\delta d_{pd}/d_{pd}^{0}\right)^{-4}$ and
$t_{pp}=t_{pp}^{0}\left(1+\delta d_{pp}/d_{pp}^{0}\right)^{-3}$, with
$d_{pd}^{0}=1.95\AA$ and $d_{pp}^{0}=\sqrt{2}d_{pd}^{0}$. An additional effect
may occur in layered structures. Liu et al. liu  showed that the presence of
Al at the interface would deplete holes on the out-of-layer oxygen sites
linking Al and Ni, raising the charge-transfer energy from those apical sites
by $\sim$1eV. We model the $M$-layer 2D structures using supercells with $M$
NiO3 units in the z-direction, terminated on both ends with apical oxygen
sites whose $e_{p}$ are shifted by -1eV. For reference we also performed some
calculations without such shift.

The interaction terms $U_{l}$ and $J_{l}$ contain the on-site repulsion and
the Hund’s interactions in the rotationally-invariant Slater-Kanamori formmit
.

$U_{l}=U\sum_{(m\sigma)>(m^{\prime}\sigma^{\prime})}n_{m\sigma}n_{m^{\prime}\sigma^{\prime}}$
(1) $\displaystyle J_{l}=$ $\displaystyle-$
$\displaystyle\frac{4}{3}j\sum_{\sigma}n_{a\sigma}n_{b\sigma}+\frac{5}{3}j\sum_{m}n_{m\uparrow}n_{m\downarrow}$
(2) $\displaystyle-$ $\displaystyle
j\sum_{\sigma}\left(\frac{1}{3}n_{a\sigma}n_{b-\sigma}+d_{a\sigma}^{\dagger}d_{b-\sigma}^{\dagger}d_{b\sigma}d_{a-\sigma}\right)$
$\displaystyle+$ $\displaystyle
j\left(d_{a\uparrow}^{\dagger}d_{a\downarrow}^{\dagger}d_{b\downarrow}d_{b\uparrow}+h.c.\right)$

In this expression, $J_{l}$ differentiates only states with the same occupancy
but inequivalent configurations.

To treat the on-site Coulomb interactions, we adopt the slave rotor (SR)
approachrotor0 ; rotor1 ; rotor2 ; rotorj0 ; rotorj1 . The method was
originally applied to the Hubbard model and then to other $d$-only models with
Hund’s-like interactionsrotorj0 ; rotorj1  under the $j<<U$ approximation. We
apply it here to the p-d model, noting that for RNiO3 j$\sim$1eV is much
smaller than either the d-d repulsion and the electron bandwidth. For each
$e_{g}$ site, the approach introduces an auxiliary SR field,
$\theta_{l}\in[0,2\pi)$, and decomposes electron operators as
$d^{\dagger}_{l\alpha\sigma}\rightarrow
f^{\dagger}_{l\alpha\sigma}e^{i\theta_{l}}$. The consistency of SR state and
d-occupancy is enforced by the constrain

$\widehat{L}_{l}=\frac{\partial}{i\partial\theta_{l}}=\sum_{\alpha\sigma}\left(f^{\dagger}_{\alpha\sigma}f_{\alpha\sigma}-\frac{1}{2}\right)$
(3)

The spectrum of $U_{l}^{(\theta)}\propto\widehat{L}_{l}^{2}$. We treat the
large $U_{l}$ with $\theta$ and the small j-scale $J_{l}$ with $f^{\dagger}$
using the weak coupling Hartree-Fock approximation. That is, we solve the
$p$-$d$ model with the single-Slater-determinant ansatz
$|MF\rangle=|p,f\rangle|\theta\rangle$. We follow previous SR applications
using Lagrange multipliers, $h_{l}$. Along with constraint Eq. 3 and up to a
constant, the system of equations reads

$\displaystyle H_{p,f}=$
$\displaystyle\sum_{l}J^{(f)}_{l}+\sum_{l\alpha\sigma}(\epsilon_{d}-\epsilon_{p}+\frac{3}{2}U-h_{l})f^{\dagger}_{l\alpha\sigma}f_{l\alpha\sigma}$
(4) $\displaystyle+$ $\displaystyle\sum_{l\epsilon\alpha\sigma}\langle
e^{-i\theta_{l}}\rangle
V_{\epsilon\alpha}p^{\dagger}_{l+\epsilon\sigma}f_{l\alpha\sigma}+h.c.$
$\displaystyle+$
$\displaystyle\sum_{l\epsilon\delta\sigma}t_{pp,\epsilon,\delta}p^{\dagger}_{l+\epsilon\sigma}p_{l+\epsilon+\delta,\sigma}+h.c.$
$\displaystyle H_{\theta}=$
$\displaystyle\sum_{l}\frac{U}{2}\widehat{L}_{l}^{2}+h_{l}\widehat{L}_{l}$ (5)
$\displaystyle+$
$\displaystyle\sum_{l}e^{i\theta_{l}}\sum_{\alpha\epsilon\sigma}V_{\epsilon\alpha}\langle
p^{\dagger}_{l+\epsilon\sigma}f_{l\sigma}\rangle+h.c.$

where $V^{0}_{\epsilon\alpha}$ is given by the product of $t^{0}_{pd}$ and the
Slater-Koster orbital symmetry factors in Table I of Ref. Slater54, .

We solve the MF equations without any restriction other than the single-
Slater-determinant assumption. A 3D Bravais lattice of Ni16O48 supercells is
required for unrestricted modeling of the $(1/2,0,1/2)$ pattern with respect
to the orthorhombic unit cell. To systematically capture the effects of
dimensionality, we model the $M$-layer structures using a 2D Bravais lattice
of Ni4MO12M+4 supercells, which is connected to the 3D Ni16O48 supercell for
$M\rightarrow\infty$. The system is solved by the standard T=0 iterative
procedure for up to 65536 supercells. Without much optimization, the worst
case initial condition with bi-partite charge-, orbital-, and magnetic-order
would converge into a PM-M or $(1/2,0,1/2)$-ordered insulator within 30 cpu
hours on an Opteron-2350 cluster.

We perform bulk calculations for (u,$e_{d}$) such that $\langle
n_{e_{g}}\rangle\sim 2$ as found in UHF with parameters fitted to
photoemission spectrumhf1 , DFTBlanca-romero11 ; Han12 ; Anisimov99 , and
DMFT+DFTPark12 . For the reference set of $U=5.3$, $j=1$,
$t_{p_{\sigma},d_{x^{2}-y^{2}}}=1.5$, $t_{pp}=0.5$,
$\Delta_{pd}=e_{d}-e_{p}+U=0.3$, the undistorted 3D lattice has $\langle
n_{e_{g}}\rangle=1.95$, slightly less than those of the above calculations. To
isolate dimensionality effects, we use the bulk’s ($u$,$e_{d}$) pairs, in
addition to the aforementioned apical $e_{p}$ shift, for layered calculations.

Figure 2:  (Left) Stable solutions’ MF energy. (Right) rotor renormalization
and insulating gap for the bulk system at $U=5.3$.

The left panel of Fig. 2 shows representative results for the energies of
different locally stable phases computed at interaction (U,j)=(5.3,1) as a
function of $(\delta d_{pd})^{2}$. We see that in agreement with experiment,
the undistorted lattice is a PM-M. As oxygen breathing disproportionation
$\delta d_{pd}$ is increased a transition to a period-4 metallic magnetic
state occurs. At yet larger distortion a metal-insulator transition occurs.
The ferromagnetic states found in UHF mizokawa ; hf1 , DFT+U Blanca-romero11 ;
Han12 ; Anisimov99  and DFT+DMFT Park12  become stable only at larger $\delta
d_{pd}$ and gain less energy than does the experimentally observed ordering
pattern. Also, $E$$\sim$$-\delta d_{pd}^{2}$ implies an equilibrium distortion
ultimately determined by the aharmonic lattice restoration force beyond the
scope of this study.

The dashed and dotted lines in the right panel of Fig. 2 are the SR
renormalizations of the p-d hopping (Eq. 4) on the two sublattices of the
distorted structure. We see that the Coulomb renormalizations are not large,
and are only weakly dependent on sublattice. This renormalization has a
moderate effect on physical properties, e.g. Fermi velocity renormalization at
(U,j)=(5.3,1) is $v_{5.3,1}/v_{0,0}\sim$0.65. The relatively modest SR
renormalizations confirms that near the “negative charge transfer” region of
the phase diagram zsa  the on-site Coulomb correlations do not drive the phase
transition in an orthodox way, in particular the physics is far from the
$\langle e^{i\theta}\rangle\rightarrow 0$ Brinkman-Rice insulator. Even then,
the renormalized kinetic terms, combined with lattice distortion and the
magnetic order opens a gap in the density of states. We stress that SR
renormalization $\langle e^{i\theta}\rangle$ due to $U$ is required for
insulation within physical parameter range.

Parameters scaled for different Re bulk materialsMedarde08 ; garcia1 ; alonso0
; alonso1  are marked on the phase diagram in top panel of Fig. 1. It captures
all experimentally observed phases over the Re series, with the exception of
Re=Pr, which is predicted to be a metal instead of an insulator, albeit with
the correct charge and magnetic order. This can be understood by noting that
our approach underestimates the insulating gap, which is 160meV at
$\frac{\delta d_{pd}}{d_{pd}^{0}}$=2.5% compared to $\sim$200meV in a 2-site
DFT+DMFT studyPark12 , and that Re=Pr is also extrapolated to be a metal from
those Re=Lu results.

In agreement with experiments, the $\Uparrow$0$\Downarrow$0 insulator (top
panel of Fig. 1) has a (1/2,0,1/2) magnetic structure with respect to the
orthorhombic unit cell, a rock-salt charge ordering pattern, but without
orbital ordering. Let us identify the Ni sites with longer/shorter Ni-O bond
length as Ni${}_{\mbox{long}}$ and Ni${}_{\mbox{short}}$ and consider the
$\frac{\delta d_{pd}}{d_{pd}^{0}}$=2.5% solution. The occupancy at those sites
are $n_{\mbox{long}}=2.03$ and $n_{\mbox{short}}$=1.90 . This
disproportionation can be understood by measuring the Ni-O hybridization order
parameter
$\widehat{O}_{l}=\sum_{\epsilon\alpha\sigma}V^{0}_{l\epsilon\alpha}e^{-i\theta_{l}}p^{\dagger}_{l+\epsilon\sigma}f_{l\alpha\sigma}+h.c.$
We found that $\frac{\langle O_{\mbox{short}}\rangle}{\langle
O_{\mbox{long}}\rangle}=1.4$, which is greater than expected from the
corresponding $\delta t_{pd}/t_{pd}^{0}\sim$10% hopping modulation. This shows
that Ni${}_{\mbox{short}}$ is strongly hybridized with the oxygen bands while
Ni${}_{\mbox{long}}$ retains its d8 characteristic. The magnetic moments are
$m_{\mbox{long}}$=1.1, which agrees with experimentalonso0 ; alonso1 , and
$m_{\mbox{short}}$=0. For charge-transfer insulators, the effective physics of
strong p-d hybridization is a strong antiferromagnetic
$\overline{S}_{d}\cdot\overline{S}_{p}$ spin correlation. In this context,
having $m_{\mbox{short}}=0$ in the single-Slater-determinant approximation is
evident of a d8-high-spin state forming a singlet with its oxygen neighbors.
The formation of $p$-$d$ singlets has also been reported in a two-site
DFT+DMFT study which predicted FM GSPark12 .

Figure 3:  Results for $u=7$. (Top) $d$-occupancy for the 1-layer structure.
(Bottom) $m_{long}$ magnetic moment for bulk, middle layer of 3-layer, 2- and
1-layer structures.

We now discuss the effects of dimensional confinement. The bulk and $M\geq 3$
layers structures have similar phase diagrams, with increasing ordered and
insulating region for decreasing $M$ and increasing $U$. In particular, the
lower panel of Fig. 3 shows that PM-M exists for $M\geq 3$. The phase diagram
of 2-, and 1-layer ultra-thin films are shown in the lower panels of Fig 1. We
see that, for $M\leq 2$ layers, the ground state is always magnetically
ordered, even in the absence of the bond disproportionation. Further, the
distortion amplitude required for the transition to the insulating phase
decreases with increasing $u$ and decreasing $M$. At (u,j)=(5.3,1), the metal-
insulator boundary $(\frac{\delta d_{pd}}{d_{pd}^{0}})_{MIT}$ is reduced from
1.9% to 0.96% and 0.44% for 2- and 1-layer, respectively. A comparison between
the solid and dotted lines in Fig. 1 shows that charge transfer to the out-of-
plane orbitalsliu  greatly enlarge the insulating regions.

To study the effects of epitaxial strain, we focus on 1-layer structure for
concreteness. We include the strain effects by scaling hopping integrals
according to the Harrison ruleharrison  for the geometry specified in Ref.
keimer, , which introduced compressive (tensile) strain with LaSrAlO4 (SrTiO3)
such that the lattice parameters are a=b=3.769 (3.853)Å and c = 3.853 (3.790)Å
. Breathing-mode distortion is then introduced as discussed above. In
agreement with experiments on ultra-thin layerskeimer ; liu1 ; moon , Fig. 1
shows that compressive (tensile) epitaxial strain enlarges (shrinks) the
insulating region. Under realistic values of $U$ and compressive stain,
ordered insulator is possible for $\delta d_{pd}$=0, agreeing with experiments
which found breathing-mode distortion only under tensile strainchakhalian .
Note that further shift in apical oxygen $e_{p}$ also decreases $\delta
d_{pd}$ required for insulation.

Figure 3 demonstrates an orbital polarization of $\sim$5%. We also found that
the polarization decreases (increases) with compressive (tensile) strain, in
agreement with a DFT+U studyHan12 ; Han11 . For example, LaSrAlO4 (SrTiO3)
substrate changes the polarization to $\sim$4 (6)%. While DFT+U predicted FM
order, our $M$=1,2 GS have an AFM in-plane order of
$\Uparrow\uparrow\Downarrow\downarrow$, which differs slightly from the bulk’s
$\Uparrow$0$\Downarrow$0 pattern by having small $m_{short}$$\neq$0 which
decreases with increasing distortion. The 2-layer structure has an additional
magnetic transition between FM and AFM ordering in the z-direction.

Lastly, we compare our results with other theoretical works. In our
formulation, the bulk PM-M develops into $\Uparrow$0$\Downarrow$0-I for
$\frac{\delta d_{pd}}{d_{pd}^{0}}>$1.9% compared to UHF’s $\frac{\delta
d_{pd}}{d_{pd}^{0}}>$7.5% transition from ferromagnetic to MCO state with
similar patternhf1 . The pattern is also similar to the $j/U>1$ S-SDW state in
the $d$-only modelbalents_prb , but we have $j/U<1/5$ and a bond and charge
disproportionation with $d^{8+\delta}\underline{L}d^{8-\delta}\underline{L}$
involving strong $pd$ hybridization. Unlike the $d$-only model, our results
are sensitive to dimensional confinement. Even though different DFT
implementations have been employed to capture the bulk LaNiO3 PM-MGou11  and
LaNiO3/LaAlO3 ordered insulating layerPuggioni12 , the demonstration of all
bulk and layered phases by a single formulation had been elusive to the best
of our knowledge.

Our formulation bridges the gap between HF-like approaches and the expensive
cluster DMFT. The results connect all ReNiO3 delicate phases in bulk and layer
form and also provide detailed insights about them. This demonstrates a viable
pathway of treating systems with different superstructures as well as
compounds such as Fe and Co oxides with strong $p$-$d$ hybridization and large
number of partially filled strongly correlated orbitals. For example, Sr2FeO4
has also been shown to exhibit strong hybridization and non-trivial magnetic
orderRozenberg98 .

We thank G. A. Sawatzky, H. Chen, H. T. Dang, R. Fernandez, S. Park, and D.
Zgid for helpful discussions. This effort is supported by US National Science
Foundation under grant NSF-DMR-1006282.

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# Classification of cyclic groups underlying only smooth skew morphisms

Kan Hu∗, István Kovács and Young Soo Kwon  K. Hu,
Department of Mathematics, Zhejiang Ocean University, Zhoushan, Zhejiang
316022, P.R. China
UP FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
UP IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
<EMAIL_ADDRESS>I. Kovács,
UP IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
UP FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
<EMAIL_ADDRESS>Y. S. Kwon,
Department of Mathematics, Yeungnam University, Kyongsan 712-749, Republic of
Korea<EMAIL_ADDRESS>
###### Abstract.

A skew morphism of a finite group $A$ is a permutation $\varphi$ of $A$ fixing
the identity element and for which there is an integer-valued function $\pi$
on $A$ such that $\varphi(ab)=\varphi(a)\varphi^{\pi(a)}(b)$ for all $a,b\in
A$. A skew morphism $\varphi$ of $A$ is smooth if the associated power
function $\pi$ is constant on the orbits of $\varphi$, that is,
$\pi(\varphi(a))\equiv\pi(a)\pmod{|\varphi|}$ for all $a\in A$. In this paper
we show that every skew morphism of a cyclic group of order $n$ is smooth if
and only if $n=2^{e}n_{1}$, where $0\leq e\leq 4$ and $n_{1}$ is an odd
square-free number. A partial solution to a similar problem on non-cyclic
abelian groups is also given.

###### Key words and phrases:

skew morphism, group factorization, solvable group

###### 2010 Mathematics Subject Classification:

05C10, 05C25, 57M15

The first and the second authors were supported by the Slovenian Research
Agency (ARRS), the first author by the research project N1-0208 and the second
author by the research program P1-0285 and research projects J1-1695, N1-0140,
J1-2451, N1-0208 and J1-3001. The third author was supported by the Basic
Science Research Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education (2018R1D1A1B05048450).
∗ Corresponding author e-mail<EMAIL_ADDRESS>
##  1\. Introduction

A _skew morphism_ of a group $A$ is a permutation $\varphi$ of $A$ fixing the
identity element of $A$ and for which there exists a function
$\pi:A\to\mathbb{Z}$ such that

$\varphi(ab)=\varphi(a)\varphi^{\pi(a)}(b)~{}\text{for all}~{}a,b\in A.$

The function $\pi$ is referred to as the _power function_ associated with
$\varphi$. If $\varphi$ is fixed, then the values of $\pi$ are uniquely
determined modulo $|\varphi|$, where $|\varphi|$ denotes the order of
$\varphi$, and therefore, $\pi$ may also be defined as a function from $A$ to
$\mathbb{Z}_{|\varphi|}$. It is a trivial observation that if $\pi(a)=1$ for
all $a\in A$, then $\varphi$ is an automorphism of $A$. Skew morphisms which
are not automorphisms are called _proper_.

The concept of a skew morphism was introduced by Jajcay and Širáň [17] to
characterize regular Cayley maps. Without going into the details, for a Cayley
map $M=\operatorname{CM}(A,X,P)$ of a group $A$, where $X$ is an inverse-
closed generating subset of $A$ and $P$ is a cyclic permutation of $X$, it was
shown that $M$ is regular if and only if $P$ extends to a skew morphism of $A$
[17, Theorem 1].

Skew morphisms are closely related to complementary factorizations of finite
groups with a cyclic factor [5]. By a _complementary factorization_ of a group
$G$ we mean that $G$ can be written as a product $AB$, where $A$ and $B$ are
subgroups of $G$ and $A\cap B=1$. Suppose that $\varphi$ is a skew morphism of
$A$. For $a\in A$, let $L_{a}$ denote the permutation of $A$ acting as
$L_{a}(x)=ax$ for all $x\in A$, and set $L_{A}=\\{L_{a}:a\in A\\}$. It is not
difficult to show that the permutation group $\langle{L_{A},\varphi}\rangle$
of $A$ admits the complementary factorization $L_{A}\langle{\varphi}\rangle$.
This group is called the _skew product group_ induced by $A$ and $\varphi$
[5].

Conversely, suppose that $G$ is any group admitting a complementary
factorization $G=AY$, where $Y$ is a cyclic subgroup and it is core-free in
$G$. Fix a generator $y$ of $Y$. Then for every $a\in A$, there is a unique
element $b\in A$ and a unique number $j\in\mathbb{Z}_{|y|}$ such that
$ya=by^{j}$. Define the mappings $\varphi:A\to A$ and
$\pi:A\to\mathbb{Z}_{|y|}$ by

$\varphi(a)=b~{}\text{and}~{}\pi(a)=j~{}\stackrel{{\scriptstyle\text{def}}}{{\iff}}~{}ya=by^{j}~{}\text{for
all}~{}a\in A.$ (1)

Then $\varphi$ and $\pi$ are well-defined, $\varphi$ is a skew-morphism of
$A$, and $\pi$ is the power function associated with $\varphi$ [5, Proposition
3.1(a)].

Regular Cayley maps, skew morphisms and skew product groups for a given
infinite family of groups have been intensively investigated. Generally
speaking, this seems a challenging problem, because even for the cyclic groups
a complete classification of the skew morphisms is not at hand, apart from the
celebrated classification of regular Cayley maps [7], and a partial
classification of the skew morphisms and the skew product groups [1, 3, 8, 14,
19, 20]. For the elementary abelian $p$-groups, a characterization of the skew
product groups and a complete classification of the regular Cayley maps can be
found in [9, 10]. For the dihedral groups, the regular Cayley maps have been
classified [18] and a characterization of the skew product groups was given
recently by the authors [13]. For the non-abelian simple groups (or more
explicitly, the monolithic groups), and non-abelian characteristically simple
groups, the skew morphisms and skew product groups have been classified, in
contrast to the fact that not much is known about the regular Cayley maps [2,
4].

In this paper we shall continue the investigation of skew morphisms of cyclic
groups. A skew morphism $\varphi$ of a group $A$ is called _smooth_ if the
associated power function $\pi$ satisfies the following condition:

$\pi(\varphi(a))\equiv\pi(a)\\!\\!\\!\\!\pmod{|\varphi|}~{}\text{for
all}~{}a\in A.$ (2)

It is clear that every automorphism is smooth, however, the converse is not
true in general. Smooth skew morphisms were defined by Hu [12] and
independently by Bachratý and Jajcay [3] under the name of _coset-preserving_
skew morphisms. The smooth skew morphisms of cyclic groups and dihedral groups
have been classified by Bachratý and Jajcay [3] (see also [15, Theorem 16] for
an alternative proof) and Wang et al. [22], respectively. To elaborate a
subtler relationship between automorphisms and smooth skew morphisms we need
to introduce the concept of a _reciprocal pair_ of skew morphisms of cyclic
groups. Suppose that $(\varphi,\tilde{\varphi})$ is a pair of skew morphisms
$\varphi$ and $\tilde{\varphi}$ of the cyclic groups $\mathbb{Z}_{m}$ and
$\mathbb{Z}_{n}$, and $\pi$ and $\tilde{\pi}$ are the associated power
functions of $\varphi$ and $\tilde{\varphi}$, respectively. Then
$(\varphi,\tilde{\varphi})$ is called _reciprocal_ if they satisfy the
following conditions:

  1. (a)

$|\varphi|$ divides $n$ and $|\tilde{\varphi}|$ divides $m$;

  2. (b)

for all $x\in\mathbb{Z}_{m}$ and $y\in\mathbb{Z}_{n}$,

$\displaystyle\pi(x)$ $\displaystyle\equiv$
$\displaystyle-\tilde{\varphi}^{-x}(-1)\\!\\!\\!\\!\pmod{|\varphi|},$
$\displaystyle\tilde{\pi}(y)$ $\displaystyle\equiv$
$\displaystyle-\varphi^{-y}(-1)\\!\\!\\!\\!\pmod{|\tilde{\varphi}|}.$

It is shown in [11, Theorem 3.5] that, for fixed $m$ and $n$, the isomorphism
classes of regular dessins with complete bipartite underlying graphs $K_{m,n}$
are in one-to-one correspondence with the reciprocal pairs
$(\varphi,\tilde{\varphi})$ of skew morphisms of $\mathbb{Z}_{m}$ and
$\mathbb{Z}_{n}$, respectively; moreover, if one of the skew morphisms is an
automorphism, then the other is necessarily smooth [15, Lemma 12].

It was shown by Kovács and Nedela [19, Theorem 6.3] that every skew morphism
of the cyclic groups $\mathbb{Z}_{n}$ is an automorphism if and only if $n=4$
or $\gcd(n,\phi(n))=1$, where $\phi$ is Euler’s totient function. Recently
Bachratý [1] showed that every skew morphism of $\mathbb{Z}_{n}$ is smooth
whenever $n=pq$, $4p$, $8p$, $16p$ or $pqr$, where $p,q,r$ are distinct
primes. Our first theorem is the following generalization.

######  Theorem 1.1.

Every skew morphism of the cyclic group $\mathbb{Z}_{n}$ is smooth if and only
if $n=2^{e}n_{1}$, where $0\leq e\leq 4$ and $n_{1}$ is an odd square-free
number.

The proof of Theorem 1.1 relies on some preliminary results from group theory
as well as the theory of skew morphisms, which will be collected in next
section.

Conder et al. [5] classified the non-cyclic abelian groups with the property
that all of their skew morphisms are automorphisms. It was shown that these
are precisely the elementary abelian $2$-groups [5, Theorem 7.5]. In this
paper we also propose the investigation of non-cyclic abelian groups with the
property that all of their skew morphisms are smooth.

######  Problem 1.

Classify the finite non-cyclic abelian groups which underly only smooth skew
morphisms.

In Section 5 we give the following partial answer.

######  Theorem 1.2.

Let $A$ be a non-cyclic abelian group of order $n=2^{f}n_{1}$, where $f\geq 0$
and $n_{1}$ is odd. If $A$ underlies only smooth skew morphisms, then $n_{1}$
is square-free, and the Sylow $2$-subgroup of $A$ contains no direct factors
isomorphic to $\mathbb{Z}_{2^{e}}$ $(e\geq 5)$.

##  2\. Preliminaries

###  2.1. Group theory

All groups in this paper will be finite. We denote the identity element of a
group $G$ by $1_{G}$ and its order by $|G|$. The order of an element $g\in G$
is denoted by $|g|$. Let $H$ be a subgroup of a group $G$. The _core of $H$ in
$G$_ is the largest normal subgroup of $G$ contained in $H$; in the case when
this is trivial, $H$ is called _core-free_ in $G$. Moreover, $C_{G}(H)$
denotes the centralizer of $H$ in $G$. If $p$ is prime, then the largest
normal $p$-subgroup of $G$ is denoted by $O_{p}(G)$. The largest normal
nilpotent subgroup of $G$ is the _Fitting subgroup_ of $G$, denoted by $F(G)$.

######  Proposition 2.1 ([21, 7.4.3, 7.4.7]).

Let $G$ be a finite group, and $\wp=\\{p_{1},\ldots,p_{k}\\}$ the set of all
prime divisors of $|G|$. Then

  1. (a)

$F(G)=\prod_{i=1}^{k}O_{p_{i}}(G)$.

  2. (b)

If $G$ is a solvable group, then $C_{G}(F(G))\leq F(G)$.

A group $G$ is called _supersolvable_ if there is a sequence

$1=N_{0}<N_{1}<\cdots<N_{k}=G$

of normal subgroups of $G$ such that $|N_{i}/N_{i-1}|$ is a prime for every
$i$, $1\leq i\leq k$. The following results are well known..

######  Proposition 2.2 ([21, 13.2.9, 13.3.1]).

Suppose that a group $G$ has a factorization $G=AB$.

  1. (a)

If both $A$ and $B$ are nilpotent, then $G$ is solvable.

  2. (b)

If both $A$ and $B$ are cyclic, then $G$ is supersolvable.

The following fact is often referred to as the Sylow tower property of
supersolvable groups.

######  Proposition 2.3 ([21, 7.2.19]).

Let $G$ be supersolvable group, and let $P_{i}$ be a Sylow $p_{i}$-subgroup of
$G$, where $\wp=\\{p_{1},p_{2},\ldots,p_{r}\\}$ constitutes the set of prime
divisors of $|G|$. If the prime divisors are ordered by $p_{i}>p_{i+1}$ for
all $i$, then for each $k$, the product $\prod_{i=1}^{k}P_{i}$ is a normal
subgroup of $G$.

Let $\wp$ be a set of primes, a positive integer $n$ is a $\wp$-number if
every prime divisor of $n$ belongs to $\wp$. If no prime divisors of $n$ lie
in $\wp$, then $n$ will be called a $\wp^{\prime}$-number. A positive divisor
$d$ of $n$ is a Hall divisor if $\gcd(d,n/d)=1$, that is, there exists some
set $\wp$ of primes such that $d$ is a $\wp$-number while $n/d$ is a
$\wp^{\prime}$-number. A group is a $\wp$-group if its order is a
$\wp$-number. A subgroup $H$ of a group $G$ is a Hall subgroup if $|H|$ is a
Hall divisor of $|G|$. Thus, $H$ is a Hall subgroup of $G$ if and only if, for
some set $\wp$ of primes, $H$ is a $\wp$-subgroup of $G$ and $|G:H|$ is a
$\wp^{\prime}$-number, in which case $H$ is also called a Hall $\wp$-subgroup
of $G$. In particular, if $\wp=\\{p\\}$ consists of a single prime, then a
Hall $\wp$-subgroup is indeed a Sylow $p$-subgroup.

######  Proposition 2.4 ([16, Kapitel VI, Satz 1.8]).

Let $G$ be a solvable group and let $\wp$ be a set consisting of some prime
divisors of $|G|$. Then the following hold:

  1. (a)

$G$ contains a Hall $\wp$-subgroup.

  2. (b)

All Hall $\wp$-subgroups are conjugate in $G$.

  3. (c)

Every $\wp$-subgroup of $G$ is contained in some Hall $\wp$-subgroup of $G$.

If $K\leq\operatorname{Aut}(G)$ and $H\leq G$, then $H$ is said to be
$K$-invariant if $\sigma(H)=H$ for every $\sigma\in K$. We shall need the
following simple lemma.

######  Lemma 2.5.

Let $N\cong\mathbb{Z}_{p}^{2}$ for a prime $p$ and let
$K\leq\operatorname{Aut}(N)$ such that $|K|$ is coprime to $p$ and $N$
contains a $K$-invariant subgroup of order $p$. Then $K$ is isomorphic to a
subgroup of $\mathbb{Z}_{p-1}^{2}$.

###### Proof.

Since $|K|$ is coprime to $p$, by Maschke Theorem (see [21, 12.1.2]) we have
$N=N_{1}\times N_{2}$, where $N_{1}$ and $N_{2}$ are $K$-invariant subgroups
of $N$ such that $N_{1}\cong N_{2}\cong\mathbb{Z}_{p}$. Thus,
$K\leq\operatorname{Aut}(N_{1})\times\operatorname{Aut}(N_{2})\cong\mathbb{Z}_{p-1}^{2}$.
∎

Suppose that $G$ acts on a finite set $\Omega$. For an element $x\in\Omega$,
we denote by $G_{x}$ the _stabilizer_ of $x$ in $G$, by
$\operatorname{Orb}_{G}(x)$ the _$G$ -orbit_ containing $x$, and by
$\operatorname{Orb}_{G}(\Omega)$ the set of all $G$-orbits. We denote by
$\operatorname{Sym}(\Omega)$ the symmetric group consisting of all permutation
of $\Omega$, and by $\mathrm{id}_{\Omega}$ the identity permutation. The
following statement is known. It will be used a couple of times in the next
section, hence we record it here.

######  Lemma 2.6.

Let $G\leq\operatorname{Sym}(\Omega)$ be a group containing a regular abelian
subgroup $A$, and suppose that $N\lhd G$. Then
$\operatorname{Orb}_{N}(\Omega)=\operatorname{Orb}_{B}(\Omega)$ for some
subgroup $B\leq A$.

###### Proof.

It is well known that $\operatorname{Orb}_{N}(\Omega)$ is a block system for
$G$. Let $K$ be the kernel of the action of $G$ on
$\operatorname{Orb}_{N}(\Omega)$. It is straightforward to check that
$\operatorname{Orb}_{K\cap A}(\Omega)=\operatorname{Orb}_{N}(\Omega)$. ∎

Let $\alpha_{i}\in\operatorname{Sym}(\Omega_{i})$, $i=1,2$. The _direct
product_ $\alpha_{1}\times\alpha_{2}$ is defined to be the permutation of
$\Omega_{1}\times\Omega_{2}$ acting as

$(\alpha_{1}\times\alpha_{2})((x_{1},x_{2}))=(\alpha_{1}(x_{1}),\alpha_{2}(x_{2}))~{}\text{for
all}~{}(x_{1},x_{2})\in\Omega_{1}\times\Omega_{2}.$

For a prime number $p$, the _affine group_ $\operatorname{AGL}(1,p)$ consists
of the permutations of the finite field $\mathbb{F}_{p}$ of the form $x\mapsto
ax+b$, where $a,b\in\mathbb{F}_{p}$ and $a\neq 0$. The following result was
known already by Galois.

######  Proposition 2.7 ([16, Kapitel II, 3.6 Satz]).

Let $G\leq\operatorname{Sym}(\Omega)$ be a transitive and solvable group and
let $|\Omega|=p$ for a prime $p$. Then $G$ is isomorphic to a subgroup of
$\operatorname{AGL}(1,p)$.

###  2.2. Skew morphisms

For a skew morphism $\varphi$ of a group $A$ with associated power function
$\pi$, it is well known that the subset

$\operatorname{Ker}\varphi:=\\{x\in A:\pi(x)\equiv
1\\!\\!\\!\\!\pmod{|\varphi|}\\}$

is subgroup of $A$, called the _kernel_ of $\varphi$ [17]. A skew morphism
$\varphi$ is _kernel-preserving_ if
$\varphi(\operatorname{Ker}\varphi)=\operatorname{Ker}\varphi$. Moreover, the
subset

$\operatorname{Core}\varphi:=\bigcap_{i=1}^{|\varphi|}\varphi^{i}(\operatorname{Ker}\varphi)$

is the core of $A$ in the skew product group $G=A\langle\varphi\rangle$ [15,
Proposition 6]. It is well known that $\varphi$ is kernel-preserving if and
only if $\operatorname{Ker}\varphi=\operatorname{Core}\varphi$. The following
properties of $\operatorname{Ker}\varphi$ is well known.

######  Proposition 2.8 ([5, 6, 17]).

Let $\varphi$ be a skew morphism of a finite group $A$ with associated power
function $\pi$. Then we have the following:

  1. (a)

For all $a,b\in A$, $\pi(a)=\pi(a)$ if and only if
$ab^{-1}\in\operatorname{Ker}\varphi$.

  2. (b)

If $A$ is an abelian group, then $\varphi$ is kernel-preserving.

  3. (c)

If $A$ is non-trivial, then the kernel $\operatorname{Ker}\varphi$ is also
non-trivial.

The index $|A:\operatorname{Ker}\varphi|$ is called the skew-type of
$\varphi$. Note that $\operatorname{Ker}\varphi=A$ if and only if $\varphi$ is
an automorphism of $A$. Using this observation and Proposition 2.8(c), we have
the following corollary.

######  Corollary 2.9.

If $p$ is a prime number, then all skew morphisms of $\mathbb{Z}_{p}$ are
automorphisms.

The direct product of skew morphisms of groups $A$ and $B$, respectively, may
not be a skew morphism of the group $A\times B$. The following criterion will
be useful.

######  Proposition 2.10 ([23]).

Let $G=A\times B$ and let $\varphi$ and $\psi$ be skew morphisms of $A$ and
$B$ with associated power functions $\pi_{\varphi}$ and $\pi_{\psi}$,
respectively.

  1. (a)

The direct product $\varphi\times\psi$ is a skew morphism of $G$ if and only
if

$\pi_{\varphi}(a)\equiv\pi_{\psi}(b)\equiv 1\\!\\!\\!\\!\pmod{d}~{}\text{for
all}~{}a\in A~{}\text{and}~{}b\in B,$

where $d=\gcd(|\varphi|,|\psi|)$.

  2. (b)

Suppose that $\varphi\times\psi$ is a skew morphism of $G$ with associated
power function $\pi$. Then for all $a\in A$ and $b\in B$,

$\pi((a,b))\equiv\pi_{\varphi}(a)\\!\\!\\!\\!\pmod{|\varphi|}\quad\text{and}\quad\pi((a,b))\equiv\pi_{\psi}(b)\\!\\!\\!\\!\pmod{|\psi|}.$

Finally, if $\varphi$ is a skew morphism of a group $A$ and $\theta$ is an
automorphism of $A$, then $\theta\varphi\theta^{-1}$ is also a skew morphism
of $A$ (see [22, Lemma 2.2]), so the automorphism group
$\operatorname{Aut}(A)$ acts on the set of skew morphisms of $A$ by
conjugation. Two skew morphisms $\varphi$ and $\psi$ of $A$ are equivalent if
they belong to the same orbit of this action.

##  3\. Constructions

In this section, for certain $n$, we construct non-smooth skew morphisms of
the cyclic group $\mathbb{Z}_{n}$. The following lemma is very useful.

######  Lemma 3.1.

Let $G=A\times B$ and let $\varphi$ and $\psi$ be skew morphisms of $A$ and
$B$, respectively. If $\varphi\times\psi$ is a skew morphism of $A\times B$,
then $\varphi\times\psi$ is smooth if and only if both $\varphi$ and $\psi$
are smooth.

###### Proof.

Let $\pi,\pi_{\varphi}$ and $\pi_{\psi}$ be the power functions associated
with $\varphi\times\psi,\varphi$ and $\psi$, respectively. Assume that
$\varphi\times\psi$ is smooth. For any $a\in A$, by Proposition 2.10(b) we
have

$\pi((\varphi\times\psi)(a,1_{B}))=\pi(a,1_{B})\equiv\pi_{\varphi}(a)\\!\\!\\!\\!\pmod{|\varphi|}$

and

$\pi((\varphi\times\psi)(a,1_{B}))=\pi(\varphi(a),1_{B})\equiv\pi_{\varphi}(\varphi(a))\\!\\!\\!\\!\pmod{|\varphi|}.$

This implies that
$\pi_{\varphi}(\varphi(a))\equiv\pi_{\varphi}(a)\\!\\!\pmod{|\varphi|}$, and
hence $\varphi$ is smooth. Similarly, $\psi$ is also smooth.

Conversely, suppose that both $\varphi$ and $\psi$ are smooth. Then for any
$(a,b)\in A\times B$, also by Proposition 2.10(b), we have

$\pi((\varphi\times\psi)(a,b))=\pi(\varphi(a),\psi(b))\equiv\pi_{\varphi}(\varphi(a))\equiv\pi_{\varphi}(a)\equiv\pi(a,b)\\!\\!\\!\\!\pmod{|\varphi|}$

and

$\pi((\varphi\times\psi)(a,b))=\pi(\varphi(a),\psi(b))\equiv\pi_{\psi}(\psi(b))\equiv\pi_{\psi}(b)\equiv\pi(a,b)\\!\\!\\!\\!\pmod{|\psi|}.$

This implies that
$\pi((\varphi\times\psi)(a,b))\equiv\pi(a,b)\\!\\!\pmod{|\varphi\times\psi|}$,
and hence $\varphi\times\psi$ is smooth. ∎

For positive integers $s$ and $t$, we define
$\tau(s,t):=\sum_{i=1}^{t}s^{i-1}.$ For integers $n$ and $r$ with $n\geq 1$,
we write $r\in\mathbb{Z}_{n}^{*}$ if $\gcd(r,n)=1$. In this case the
_multiplicative order_ of $r$ in $\mathbb{Z}_{n}$ is defined to be the
smallest positive integer $l$ for which $r^{l}\equiv 1\\!\\!\pmod{n}$.

######  Proposition 3.2 ([3, 15]).

For $n>1$, the proper smooth skew morphisms of $\mathbb{Z}_{n}$ and the
associated power functions are given by the formula

$\displaystyle\varphi(x)\equiv
x+rk\frac{\tau(s,t)^{x}-1}{\tau(s,t)-1}\\!\\!\\!\\!\pmod{n}~{}\text{and}~{}\pi(x)\equiv
t^{x}\\!\\!\\!\\!\pmod{m},$ (3)

where $k>1$ is a proper divisor of $n$, $r\in\mathbb{Z}_{n/k}$,
$s\in\mathbb{Z}_{n/k}^{*}$ and $t\in\mathbb{Z}_{m}^{*}$ are positive integers
satisfying the following conditions:

  1. (a)

$m$ is the smallest positive integer such that $r\sum_{i=1}^{m}s^{i-1}\equiv
0\\!\\!\pmod{n/k}$.

  2. (b)

$t$ has multiplicative order $k$ in $\mathbb{Z}_{m}$.

  3. (c)

$s-1\equiv
r\big{(}(\sum_{i=1}^{t}s^{i-1})^{k}-1\big{)}/(\sum_{i=1}^{t}s^{i-1}-1)\\!\\!\pmod{n/k}$.

  4. (d)

$s^{t-1}\equiv 1\\!\\!\pmod{n/k}$.

Moreover, $k$ is equal to the skew-type of $\varphi$, and $m$ is equal to the
order of $\varphi$.

We remark that, even though all smooth skew morphisms of $\mathbb{Z}_{n}$ are
determined, it is not clear from Proposition 3.2 that, for which $n$, the
group $\mathbb{Z}_{n}$ underlies a non-smooth skew morphism. We also observe
in (3) that the proper smooth skew morphisms of $\mathbb{Z}_{n}$ are all
defined by exponential functions on $\mathbb{Z}_{n}$. By contrast, it is well
known that every automorphism of $\mathbb{Z}_{n}$ is a linear function of the
form $x\mapsto rx$, where $r\in\mathbb{Z}_{n}^{*}$. Surprisingly enough, it
was shown in [14] that the proper skew morphisms of $\mathbb{Z}_{n}$ which are
square roots of automorphisms are quadratic polynomials over the ring
$\mathbb{Z}_{n}$.

######  Proposition 3.3 ([14]).

Every proper skew morphism $\varphi$ of $\mathbb{Z}_{n}$ such that
$\varphi^{2}$ is an automorphism of $\mathbb{Z}_{n}$ is equivalent to a skew
morphism of the form

$\varphi(x)\equiv sx-\frac{x(x-1)n}{2k}\\!\\!\\!\\!\pmod{n},$

where $k$ and $s$ are positive integers satisfying the following conditions:

  1. (a)

$k^{2}$ divides $n$ and $s\in\mathbb{Z}_{n}^{*}$ if $k$ is odd, and $2k^{2}$
divides $n$ and $s\in\mathbb{Z}_{n/2}^{*}$ if $k$ is even.

  2. (b)

$s\equiv-1\\!\\!\pmod{k}$, $s$ has multiplicative order $2\ell$ in
$\mathbb{Z}_{n/k}$, and $\gcd(w,k)=1$, where

$w\equiv\frac{k}{n}(s^{2\ell}-1)-\frac{s(s-1)}{2}\ell\\!\\!\\!\\!\pmod{k}.$

The skew-type of $\varphi$ is equal to $k$, and the order of $\varphi$ is
equal to $m:=2k\ell$, and finally, the power function of $\varphi$ is given by

$\pi(x)\equiv 1+2xw^{\prime}\ell\\!\\!\\!\\!\pmod{m},$

where $w^{\prime}$ is determined by the congruence $w^{\prime}w\equiv
1\\!\\!\pmod{k}$.

We are ready to show that $\mathbb{Z}_{n}$ underlies a non-smooth skew
morphism for certain $n$.

######  Lemma 3.4.

  1. (a)

For any $e\geq 2$, the cyclic group $\mathbb{Z}_{p^{e}}$ underlies a non-
smooth skew morphism, where $p$ is an odd prime.

  2. (b)

For any $e\geq 5$, the cyclic group $\mathbb{Z}_{2^{e}}$ underlies a non-
smooth skew morphism.

###### Proof.

(a) In Proposition 3.3, take $n:=p^{e}$, where $e\geq 2$, and set $k:=p$,
$s:=-1$. Then $\ell=1$, $w=w^{\prime}=-1$ and $m=2p$. It is easy to verify
that the stated conditions are satisfied, so we obtain a skew-morphism of
$\mathbb{Z}_{p^{e}}$ and the associated power function given by

$\varphi(x)\equiv-
x-p^{e-1}\frac{x(x-1)}{2}\\!\\!\\!\\!\pmod{p^{e}}\quad\text{and}\quad\pi(x)\equiv
1-2x\\!\\!\\!\\!\pmod{2p}.$

Since $\pi(\varphi(1))\equiv\pi(-1)\equiv 3\\!\\!\pmod{2p}$ and
$\pi(1)\equiv-1\\!\\!\pmod{2p}$, we have
$\pi(\varphi(1))\not\equiv\pi(1)\\!\\!\pmod{2p}$, so $\varphi$ is not smooth.

(b) In Proposition 3.3, let $n:=2^{e}$ where $e\geq 5$, and take $k:=4$ and
$s:=-1$. As before we have $\ell=1$, $w=w^{\prime}=-1$ and $m=8$. It is easy
to verify that the stated conditions are satisfied, so we obtain a skew
morphism of $\mathbb{Z}_{2^{e}}$ and the associated power function given by

$\varphi(x)=-x-2^{e-3}x(x-1)\\!\\!\\!\\!\pmod{2^{e}}\quad\text{and}\quad\pi(x)\equiv
1-2x\\!\\!\\!\\!\pmod{8}.$

Since $\pi(\varphi(1))\equiv\pi(-1)\equiv 3\\!\\!\pmod{8}$ and
$\pi(1)\equiv-1\\!\\!\pmod{8}$,
$\pi(1)\not\equiv\pi(\varphi(1))\\!\\!\pmod{8}$, so $\varphi$ is also not
smooth. ∎

######  Lemma 3.5.

Let $n$ be a positive integer with a decomposition $n=2^{e}n_{1}$, where
$n_{1}$ is odd. If $e\geq 5$ or $n_{1}$ is not square-free, then
$\mathbb{Z}_{n}$ underlies a non-smooth skew morphism.

###### Proof.

Note that $\mathbb{Z}_{n}\cong\mathbb{Z}_{2^{e}}\times\mathbb{Z}_{n_{1}}$. If
$e\geq 5$, then by Lemma 3.4(b), the cyclic group $\mathbb{Z}_{2^{e}}$ has a
non-smooth skew morphism $\alpha$, so by Lemma 3.1,
$\varphi:=\alpha\times\mathrm{id}_{n_{1}}$ is a non-smooth skew morphism of
$\mathbb{Z}_{n}$, where $\operatorname{id}_{n_{1}}$ denotes the identity
permutation on $\mathbb{Z}_{n_{1}}$. If $n_{1}$ is not square-free, then there
is an odd prime $p$ such that $p^{e}\mid n_{1}$ where $e\geq 2$, using Lemma
3.4(a) and similar techniques we can construct a non-smooth skew morphism of
$\mathbb{Z}_{n}$, as required. ∎

##  4\. Proof of Theorem 1.1

The aim of this section is to prove Theorem 1.1. Before doing this, we make
the following convention. Suppose that $G=AY$ is a complementary
factorization, where the subgroup $Y$ is cyclic and core-free in $G$. Let $y$
be a fixed generator of $Y$, then a skew morphism $\varphi$ of $A$ together
with the associated power function $\pi$ are defined according to (1). The
skew morphism $\varphi$ will be referred to as the _skew morphism of $A$
induced by $y$_. Note that the action of $G$ on the set $\Omega:=\\{gY\mid
g\in G\\}$ of left cosets of $Y$ in $G$ is faithful; in particular, the
subgroup $A$ acts regularly on $\Omega$ and $Y$ is a point stabilizer.
Therefore, we may assume that $G\leq\operatorname{Sym}(\Omega)$ with $G=AY$,
where $A$ is a regular subgroup of $G$ and $Y=G_{x}$ is cyclic for some
$x\in\Omega$. This assumption will be used throughout this section and for the
sake of convenience we record it here.

######  Assumption 4.1.

Let $G\leq\operatorname{Sym}(\Omega)$ be a group such that $G=AY$, where $A$
is abelian and regular on $\Omega$, $Y=G_{x}=\langle{y}\rangle$ for a fixed
element $x\in\Omega$, and let $\varphi$ be the skew morphism induced by $y$.

The proof of Theorem 1.1 will be given after four preparatory lemmas.

######  Lemma 4.2.

With the notations in Assumption 4.1, suppose that $O_{p}(G)\leq A$ for every
prime divisor $p$ of $|G|$. Then $A\lhd G$.

###### Proof.

Denote by $\wp=\\{p_{1},p_{2},\ldots,p_{k}\\}$ the set of prime divisors of
$|G|$, then by Proposition 2.1(a) $F(G)=\prod_{i=1}^{k}O_{p_{i}}(G)\leq A$. On
the other hand, by Proposition 2.2(a) $G$ is solvable, and by Proposition
2.1(b) we have $A\leq C_{G}(F(G))\leq F(G)$. Therefore $A=F(G)$, in
particular, $A\lhd G$. ∎

For a prime divisor $p$ of $|A|$, we denote by $A_{p}$ the (unique) Sylow
$p$-subgroup of the abelian group $A$ and by $A_{p^{\prime}}$ the (unique)
Hall $p^{\prime}$-subgroup of $A$.

######  Lemma 4.3.

With the notations in Assumption 4.1, suppose that $O_{p}(G)\neq 1$ for some
prime divisor $p$ of $|A|$ and $|A_{p}|=p$. Then $O_{p}(G)=A_{p}$ or
$A_{p}\times Y^{*}$, where $Y^{*}\leq Y$ and $|Y^{*}|=p$.

###### Proof.

Write $N:=O_{p}(G)$. By Lemma 2.6 we know that
$\operatorname{Orb}_{N}(\Omega)=\operatorname{Orb}_{B}(\Omega)$ for some
$B\leq A$. Since the size of each $N$-orbit is a power of $P$, we have
$B=A_{p}$. Thus, $\operatorname{Orb}_{N}(\Omega)$ is a block system for $G$ in
which each block has size $p$. Let $K$ be the kernel of the action of $G$ on
$\operatorname{Orb}_{N}(\Omega)$. By Proposition 2.2(a) $G$ is solvable, so
$K$ is solvable as well. It follows from Proposition 2.7 that $K$ is
isomorphic to a subgroup of the direct product
$\operatorname{AGL}(1,p)\times\cdots\times\operatorname{AGL}(1,p)$ with
$|A|/p$ factors. It is clear that the Sylow $p$-subgroup $P$ of $K$ is normal
in $K$ and it is elementary abelian. Thus, $P\operatorname{~{}char~{}}K\lhd G$
and hence $P\lhd G$, we obtain $P\leq O_{p}(G)$. On the other hand, since
$O_{p}(G)=N\leq K$, we also have $O_{p}(G)\leq P$. Therefore, $O_{p}(G)=P$,
and we conclude that $O_{p}(G)=A_{p}$ or $A_{p}\times Y^{*}$, as required. ∎

######  Lemma 4.4.

With the notations in Assumption 4.1, suppose that $N$ is a normal subgroup of
$G$. For $g\in G$, denote by $\bar{g}$ the image of $g$ under its action on
$\operatorname{Orb}_{N}(\Omega)$ and for $H\leq G$, set
$\bar{H}=\\{\bar{h}:h\in H\\}$.

  1. (a)

$\bar{G}=\bar{A}\bar{Y}$, $\bar{A}\cap\bar{Y}=1$ and $\bar{Y}$ is core-free in
$\bar{G}$.

  2. (b)

Let $\bar{\varphi}$ be the skew morphism of $\bar{A}$ induced by $\bar{y}$.
Then

$\bar{\varphi}(\bar{a})=\overline{(\varphi(a))}~{}\text{for all}~{}a\in A.$

  3. (c)

Let $\bar{\pi}$ be a power function associated with $\bar{\varphi}$. Then

$\bar{\pi}(\bar{a})\equiv\pi(a)\\!\\!\\!\\!\pmod{|\bar{\varphi}|}~{}\text{for
all}~{}a\in A.$

###### Proof.

The mapping $g\mapsto\bar{g}$ is an epimorphism from $G$ onto $\bar{G}$,
therefore, $\bar{G}=\bar{A}\bar{Y}$. Since $\bar{A}$ is abelian and transitive
on $\operatorname{Orb}_{N}(\Omega)$, it is regular. It is evident that
$\bar{Y}$ is contained in the stabilizer of the $N$-orbit containing the
identity element $1_{A}$. Denoting this stabilizer by $Z$, we have
$\bar{A}\bar{Y}=\bar{G}=\bar{A}Z$. Since $\bar{A}$ is regular, we have
$\bar{A}\cap\bar{Y}=\bar{1}=\bar{A}\cap Z$, so $Z=\bar{Y}$. This proves (a).

Fix any $a\in A$, we have $ya=\varphi(a)y^{\pi(a)}$, so
$\bar{y}\bar{a}=\overline{\varphi(a)}(\bar{y})^{\pi(a)}$. Then (1) can be
applied to the factorisation $\bar{A}\bar{Y}$ and generator
$\bar{y}\in\bar{Y}$, giving rise to the equality
$\bar{\varphi}(\bar{a})=\overline{\varphi(a)}$ and the congruence
$\bar{\pi}(\bar{a})\equiv\pi(a)\pmod{|\bar{\varphi}|}$, this proves (b) and
(c). ∎

In what follows, the skew morphism $\bar{\varphi}$ defined as in the lemma
above will be called _the quotient of $\varphi$ determined by $N$_, and
denoted by $\varphi_{N}$.

######  Lemma 4.5.

With the notations in Assumption 4.1, suppose that $G$ has two distinct normal
subgroups $M$ and $N$ satisfying the following conditions:

  1. (a)

There are subgroups $B,\,C\leq A$ such that $B\cap C=1$, and

$\operatorname{Orb}_{M}(\Omega)=\operatorname{Orb}_{B}(\Omega)~{}\text{and}~{}\operatorname{Orb}_{N}(\Omega)=\operatorname{Orb}_{C}(\Omega).$

  2. (b)

Both skew morphisms $\varphi_{M}$ and $\varphi_{N}$ are smooth.

Then $\varphi$ is smooth.

###### Proof.

For any $a\in A$, since $\varphi_{M}$ is smooth, by Lemma 4.4(a)–(b), we have

$\pi(\varphi(a))\equiv\bar{\pi}(\overline{\varphi(a)})=\bar{\pi}(\varphi_{M}(\bar{a}))\equiv\bar{\pi}(\bar{a})\equiv\pi(a)\\!\\!\\!\\!\pmod{|\varphi_{M}|}.$

Similarly, $\pi(\varphi(a))\equiv\pi(a)\\!\\!\pmod{|\varphi_{N}|}.$

In what follows we show that
$|\varphi|=\operatorname{lcm}(|\varphi_{M}|,|\varphi_{N}|)$, which together
with the above congruences will imply
$\pi(\varphi(a))\equiv\pi(a)\\!\\!\pmod{|\varphi|}$ for all $a\in A$, and
hence $\varphi$ is smooth. Indeed, let $K$ and $L$ be the kernels of the
actions of $G$ on $\operatorname{Orb}_{M}(\Omega)$ and
$\operatorname{Orb}_{N}(\Omega)$, respectively. Then

$|Y|=|\varphi_{M}||Y\cap K|=|\varphi_{N}||Y\cap L|.$ (4)

Let $d=\gcd(|\varphi_{M}|,|\varphi_{N}|)$. Now by (4) we have

$\frac{|\varphi_{M}|}{d}\cdot|Y\cap K|=\frac{|\varphi_{N}|}{d}\cdot|Y\cap L|.$
(5)

If $\psi\in(Y\cap K)\cap(Y\cap L)$, then for any $x\in\Omega$, $\psi$ fixes
the orbits $\operatorname{Orb}_{M}(x)$ and $\operatorname{Orb}_{N}(x)$, so by
(a) we have
$\psi(x)\in\operatorname{Orb}_{B}(x)\cap\operatorname{Orb}_{C}(x)$. Since
$B\cap C=1$ and $A$ is regular, we have
$\operatorname{Orb}_{B}(x)\cap\operatorname{Orb}_{C}(x)=\\{x\\}$, and so
$\psi=\operatorname{id}_{\Omega}$. But $Y$ is a cyclic group, we obtain
$\gcd(|Y\cap K|,|Y\cap L|)=1$, and from (5) we deduce that $|Y\cap K|$ divides
$|\varphi_{N}|/d$. On the other hand, $|\varphi_{N}|/d$ divides $|Y\cap K|$
because it is coprime to $|\varphi_{M}|/d$, and we find $|Y\cap
K|=|\varphi_{N}|/d$. Thus, by (4)
$|\varphi|=|Y|=\operatorname{lcm}(|\varphi_{M}|,|\varphi_{N}|).$ ∎

###### Proof of Theorem 1.1.

We keep the notations set in Assumption 4.1 and, in addition, assume that
$A\cong\mathbb{Z}_{n}$ for some $n\geq 1$. We have to show that $\varphi$ is
smooth if and only if $n=2^{e}n_{1}$, where $0\leq e\leq 4$ and $n_{1}$ is an
odd square-free number. The necessity has been proved in Lemma 3.5. For the
converse, we proceed by induction on $n$. If $n=2^{e}$ $(0\leq e\leq 4)$, then
by the census in [1] we know that $\varphi$ is smooth; if $n$ is an odd prime,
then $\varphi$ is an automorphism of $A$ due to Corollary 2.9, in particular,
it is smooth.

From now on we will assume that $n$ is either an odd composite number or an
even number with an odd prime divisor and also that the theorem holds for any
cyclic group whose order is a proper divisor of $n$. By Proposition 2.2(b),
$G$ is supersolvable, and therefore, Proposition 2.3 can be applied to $G$. In
particular, if $p$ is the largest prime divisor of $|G|$, then the Sylow
$p$-subgroup $P$ of $G$ is normal in $G$. If $O_{q}(G)\neq 1$ for another
prime divisor $q$ of $|G|$, then Lemma 4.5 can be applied to $G$ with $M=P$
and $N=O_{q}(G)$. Using also the induction hypothesis, we obtain that
$\varphi$ is smooth.

Thus we may assume that $F(G)=O_{p}(G)=P$. Notice that $p>2$, so $|A_{p}|=p$.
By Lemma 4.3 we have $P=A_{p}$ or $P=A_{p}\times
Y^{*}\cong\mathbb{Z}_{p}^{2}$, where $Y^{*}\leq Y$ and $|Y^{*}|=p$. If
$P=A_{p}$, then from Lemma 4.2 we deduce that $A\lhd G$, and so $\varphi$ is
an automorphism; in particular, $\varphi$ is smooth. In what follows we assume
$P=A_{p}\times Y^{*}$.

By Proposition 2.4 we may assume that $Q$ is a Hall $p^{\prime}$-subgroup of
$G$ containing $A_{p^{\prime}}$. Then $G=P\rtimes Q$ and by Proposition 2.1
$C_{G}(P)=C_{G}(F(G))=F(G)=P$. It follows that $Q$ is isomorphic to a subgroup
of $\operatorname{Aut}(P)$. By Proposition 2.8(b)–(c),
$1\neq\operatorname{Ker}\varphi\lhd G$, and so
$1\neq\operatorname{Ker}\varphi\leq F(G)\cap A=A_{p}$. But $|A_{p}|=p$, we get
$A_{p}=\operatorname{Ker}\varphi$ and $A_{p}$ is a normal subgroup of $G$.
Thus, $A_{p}$ is $Q$-invariant. Since $p\nmid|Q|$, by Lemma 2.5 the subgroup
$Q$ is isomorphic to a subgroup of $\mathbb{Z}_{p-1}^{2}$, so it is abelian.
Now $Q\cong G/P\cong\bar{G}$, where $\bar{G}$ is the image of $G$ induced by
its action on $\operatorname{Orb}_{P}(\Omega)$. By Lemma 4.4
$Q\cong\bar{A}\bar{Y}$ and the latter group is the skew product group induced
by $\bar{A}$ and the quotient skew morphism $\bar{\varphi}$. Since it is
abelian, $\bar{Y}=1$, so $Y\leq P$. Therefore, $|\varphi|=|Y|=p$, and hence
$\varphi$ is smooth. ∎

##  5\. non-cyclic abelian groups

In this section, on the basis of the results obtained in the previous
sections, we shall give a partial solution to Problem 1. The following
classification of proper skew morphisms of the elementary abelian group
$\mathbb{Z}_{p}\times\mathbb{Z}_{p}$ will be useful.

######  Proposition 5.1 ([5, Theorem 5.10]).

Let $(a,x)$ be a basis of the elementary abelian group
$A\cong\mathbb{Z}_{p}\times\mathbb{Z}_{p}$, where $p$ is an odd prime. Then
every proper skew morphisms of $A$ can be expressed as the form

$\varphi(a^{i}x^{j})=a^{ri+\frac{1}{2}dj(j-1)rn}(bx^{r})^{j},$

where $b$ is an element in the kernel $\operatorname{Ker}\varphi=\langle
a\rangle$ uniquely determined by the triple $(d,n,r)$, for all
$d,n\in\mathbb{Z}_{p}^{*}$ and $r\in\\{2,\ldots,p-1\\}$. The skew morphism
$\varphi$ has order $pk$, where $k$ is the multiplicative order of $r$ modulo
$p$, and its power function is given by

$\quad\pi(a^{i}x^{j})\equiv 1+jnk\\!\\!\\!\\!\pmod{pk}.$

######  Lemma 5.2.

Let $p$ be an odd prime, then every proper skew morphism of the elementary
abelian group $\mathbb{Z}_{p}\times\mathbb{Z}_{p}$ is non-smooth.

###### Proof.

Using the notation in Proposition 5.1, we have $\pi(x)\equiv
1+nk\\!\\!\pmod{pk}$ and
$\pi(\varphi(x))\equiv\pi(bx^{r})\equiv\pi(x^{r})\equiv 1+rnk\\!\\!\pmod{pk}$,
and so $\pi(x)\not\equiv\pi(\varphi(x))\\!\\!\pmod{pk}$. Therefore, $\varphi$
is not smooth. ∎

######  Lemma 5.3.

Let $G=A\times B$ be an abelian group such that
$A\cong\mathbb{Z}_{p}\times\mathbb{Z}_{p}$, where $p$ is an odd prime. Then
$G$ underlies a non-smooth skew morphism.

###### Proof.

By Lemma 5.1, the elementary abelian group
$A\cong\mathbb{Z}_{p}\times\mathbb{Z}_{p}$ has a non-smooth skew morphism
$\alpha$, so by Lemma 3.1 $\varphi:=\alpha\times\mathrm{id}_{B}$ is a non-
smooth skew morphism of $G$. ∎

###### Proof of Theorem 1.2.

If the Sylow $2$-group $P$ of $A$ contains a direct factor isomorphic to
$\mathbb{Z}_{2^{e}}$ for some $e\geq 5$, then by Lemma 3.1 and Lemma 3.4 we
can construct a non-smooth skew morphism of $A$.

If $n_{1}$ is not square-free. Then there is an odd prime $p$ such that
$p^{2}\mid n$, and so either $A\cong\mathbb{Z}_{p}\times\mathbb{Z}_{p}\times
B$, or $A\cong\mathbb{Z}_{p^{i}}\times B$ for some $i\geq 2$ and some subgroup
$B\leq A$. By Lemmas 3.4(a) and 5.3, both $\mathbb{Z}_{p}\times\mathbb{Z}_{p}$
and $\mathbb{Z}_{p^{i}}$ underly a non-smooth skew morphism $\alpha$.
Therefore, by Lemma 3.1 the skew morphism
$\varphi=\alpha\times\operatorname{id}_{B}$ is a non-smooth skew morphism of
$A$. ∎

Let $A$ be a non-cyclic abelian group of order $n=2^{f}n_{1}$, where $n_{1}$
is odd. If every skew morphism of $A$ is smooth. Then, by Theorem 1.2, $n_{1}$
is square-free and the Sylow $2$-subgroup $P$ of $A$ contains no direct
factors isomorphic to $\mathbb{Z}_{2^{e}}$ for any $e\geq 5$. Thus,
$P\cong\mathbb{Z}_{2^{e_{1}}}\times\cdots\times\mathbb{Z}_{2^{e_{r}}}$ for
some positive integer $r$, where $1\leq e_{i}\leq 4$ for all $1\leq i\leq r$.
In particular, if $e_{1}=e_{2}=\cdots=e_{r}=1$, it is known that every skew
morphism of $P$ is an automorphism (see [5, Theorem 5.8]), so it is smooth.
But for other cases, little is known. For an abelian $2$-group
$\mathbb{Z}_{2^{e_{1}}}\times\cdots\times\mathbb{Z}_{2^{e_{r}}}$, we call the
$r$-tuple $(e_{1},\ldots,e_{r})$ admissible if every skew morphism of the
abelian group $\mathbb{Z}_{2^{e_{1}}}\times\cdots\times\mathbb{Z}_{2^{e_{r}}}$
is smooth.

######  Problem 2.

Determine the admissible $r$-tuples for all $r\geq 1$.

######  Problem 3.

Suppose that $(e_{1},\ldots,e_{r})$ is an admissible $r$-tuple, and $A$ is an
abelian group such that its Sylow $2$-subgroup
$P\cong\mathbb{Z}_{2^{e_{1}}}\times\cdots\times\mathbb{Z}_{e^{r}}$ and the
order of its Hall $2^{\prime}$-subgroup is square-free, is it true that every
skew morphism of $A$ is smooth?

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