Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 4 (2011) 329–349
Decomposition of skew-morphisms of cyclic
groups
István Kovács ∗
FAMNIT, University of Primorska, Glagoljaška 8, 6000 Koper, Slovenia
Roman Nedela †
Institute of Mathematics, Slovak Academy of Science, Ďumbierska 1, 975 49 Banská
Bystrica, Slovakia
Received 10 December 2009, accepted 16 July 2011, published online 1 October 2011
Abstract
A skew-morphism of a group H is a permutation σ of its elements fixing the identity
such that for every x, y ∈ H there exists an integer k such that σ(xy) = σ(x)σk(y). It
follows that group automorphisms are particular skew-morphisms. Skew-morphisms ap-
pear naturally in investigations of maps on surfaces with high degree of symmetry, namely,
they are closely related to regular Cayley maps and to regular embeddings of the complete
bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in
the context of the associated Schur rings. We prove the following decomposition theorem
about skew-morphisms of cyclic groups Zn: if n = n1n2 such that (n1, n2) = 1, and
(n1, φ(n2)) = (φ(n1), n2) = 1 (φ denotes Euler’s function) then all skew-morphisms σ of
Zn are obtained as σ = σ1×σ2, where σi are skew-morphisms of Zni , i = 1, 2. As a con-
sequence we obtain the following result: All skew-morphisms of Zn are automorphisms of
Zn if and only if n = 4 or (n, φ(n)) = 1.
Keywords: Cyclic group, permutation group, skew-morphism, Schur ring.
Math. Subj. Class.: 05C25, 05E18
1 Introduction
Let G be a group with identity element e. A nonidentity permutation σ of G with finite
order ord(σ) = m is said to be a skew-morphism of G if there exists a function π from G
to {1, . . . ,m} such that
∗Supported in part by “ARRS – Agencija za raziskovanje Republike Slovenije”, program no. P1-0285.
†Suported by the Slovak Ministry of Education, grant VEGA 2/5132/26.
E-mail addresses: istvan.kovacs@upr.si (István Kovács), nedela@savbb.sk (Roman Nedela)
Copyright c© 2011 DMFA Slovenije
330 Ars Math. Contemp. 4 (2011) 329–349
1. σ(e) = e, and
2. σ(xy) = σ(x)σπ(x)(y) for all x, y ∈ G.
The function π is called the power function of σ. For example, all automorphisms of
G are skew-morphisms, and these can be characterised as those having power function
π(x) = 1 for all x ∈ G. One can prove that the above definition of skew-morphism is
equivalent to the one given in the abstract. We adopt the terminology pure skew-morphism
for a skew-morphism which is not an automorphism of G. Following [7], we denote by
Skew(G) the set of all skew-morphisms of G.
The notion of a skew-morphism was introduced by Jajcay and Širáň in [7] in the context
of regular Cayley maps. Let G be a finite group and ρ be a cyclic permutation of a set
of generators X of G closed under taking inverses. The pair (G, ρ) determines a 2-cell
embedding of the (right) Cayley graph Cay(G;X) into an orientable surface, where the
cyclic permutation of arcs based at a vertex is induced by ρ and G. The corresponding
map is called the Cayley map CM(G; ρ). By definition, the group of left translations GL
acts as a group of map automorphisms of CM(G; ρ). It follows that Cayley maps are
vertex-transitive. Jajcay and Širáň proved that the map automorphism group of CM(G; ρ)
is regular on arcs if and only if ρ extends to a skew-morphism σ of G. Moreover, the map
automorphism group of the regular Cayley map CM(G; ρ) is a product GL〈σ〉. For more
details we refer the reader to the papers [3, 4, 7]. In particular, an ongoing project of Conder
and Tucker is aimed towards a classification of all regular Cayley maps coming from cyclic
groups, equivalently, this requires to understand skew-morphisms of Zn admitting an orbit
T for which T = T−1 and T generates Zn. Note that there are skew-morphisms of Zn with
no such orbit. Skew-morphisms of cyclic groups appear in the study of regular embeddings
of complete bipartite graphs as well. More precisely, a classification of regular embeddings
of Kn,n is equivalent to the description of the set of all skew-morphisms σ of Zn whose
order divides n, and whose power function π satisfies π(x) = −σ−x(−1) for all x ∈ Zn
(see [14, 12]). Skew-morphisms of Zn posessing these properties are called n-admissible
in [12]. A classification of regular embeddings of Kn,n was completed recently by Jones
[8]; this result was preceeded by several papers treating partial cases (see [5, 6, 9, 10]). In
particular, a formula enumerating all n-admissible skew-morphisms in Skew(Zn) can be
found in [8].
In this paper we consider skew-morphisms of Zn with no restriction. The set of all
skew-morphisms of Zn will be denoted by Skew(Zn). For i = 1, 2, let σi be a permutation
of Zni . The direct product σ1 × σ2 is the permutation of Zn1 × Zn2 defined as (σ1 ×
σ2) : (x1, x2) 7→ (σ1(x1), σ2(x2)) for all (x1, x2) ∈ Zn1 × Zn2 . Notice that if σi is a
skew-morphism of Zni for both i = 1, 2, σ1 × σ2 is not necessarily a skew-morphism of
Zn1 × Zn2 . In what follows φ denotes Euler’s function.
In this paper we prove the following decomposition theorem.
Theorem 1.1. Let n be a natural number which admits a decomposition n = n1n2, where
(n1, n2) = 1 and (n1, φ(n2)) = (φ(n1), n2) = 1. Then σ ∈ Skew(Zn) if and only if
σ = σ1 × σ2, and σi ∈ Skew(Zni) for i = 1, 2.
In proving Theorem 1.1 our main tool is to study the permutation groupG = 〈(Zn)L, σ〉
≤ Sym(Zn), where (Zn)L is the left regular representation of Zn. In particular, we study
the transitivity module over Zn induced by 〈σ〉, to which we shall also refer as the S-ring
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 331
generated by σ, and denote it by V (σ). At this point we remark that, the transitivity mod-
ules over Zn which are induced by cyclic permutation groups with a faithful orbit are,
in general, not known. To give a description of these S-rings seems to be an interesting
problem in the theory of S-rings over cyclic groups.
An outline of the paper is as follows. Section 2 contains definitions and facts on skew-
morphisms and S-rings needed for further use. In Sections 3-5 we consider the S-rings
V (σ), σ ∈ Skew(Zn). In Section 3 we apply the structure theorems of Evdokimov and
Ponomarenko to the S-rings V (σ) and discuss some consequences for the skew-morphisms
σ. In Sections 4 and 5 we restrict to special orders n. In Section 4 we set n = pe, p is
an odd prime, and prove that the S-rings V (σ) in this case coincide with those generated
by group automorphisms (see Theorem 4.1). This result is used to construct the whole set
Skew(Zp2). In Section 5 we set n = pq, where p, q are distinct primes. We first determine
all S-rings V (σ), and using this result we construct the whole set Skew(Zpq). Theorem 1.1
is proved in Section 6.
2 Preliminaries
1. Skew-morphisms
First we summarize some properties of skew-morphisms proved in [7].
Proposition 2.1. Let σ be a skew-morphism of a group H and let π be the power function
of σ. Then the following hold.
(i) For all k ≥ 1, and all x, y ∈ H , σk(xy) = σk(x)σ
∑k−1
i=0 π(σ
i(x))(y).
(ii) The set kerπ = {x ∈ H : π(x) = 1} is a subgroup of H .
(iii) For all x, y ∈ G, π(x) = π(y) if and only if x and y belong to the same right coset of
the subgroup kerπ in H .
For an arbitrary group H and h ∈ H , let hL be the permutation in Sym(H) defined
by hL(x) = hx, x ∈ H . Thus HL = {hL | h ∈ H} is the left regular representation
of H . Throughout the paper a permutation group G ≤ Sym(H) with HL ≤ G will be
referred to as an overgroup of HL in Sym(H). Skew-morphisms of H appear naturally
in the investigation of overgroups of HL in Sym(H) with cyclic point stabilizers. The
following proposition relates skew-morphisms of groups and products of groups, where
one of the factors is cyclic.
Proposition 2.2. Let G be an overgroup of HL in Sym(H), and Ge be the stabilizer
of e in G. If Ge is a cyclic group, then any generator σ of Ge is a skew-morphism of
H . Conversely, if σ is a skew-morphism of a group H , then the permutation group G =
〈HL, σ〉 has point stabilizer Ge = 〈σ〉.
Proof. Let Ge = 〈σ〉. By the assumptions G is a product G = GeHL = HLGe. It follows
that for every x ∈ G there exists a unique integer k, 1 ≤ k < ord(σ), and a unique y ∈ H
such that σxL = yLσk. Setting y = σ(x) and k = π(x) we get the identity
σxL = σ(x)Lσ
π(x). (2.1)
332 Ars Math. Contemp. 4 (2011) 329–349
Let us fix x and apply the above permutations to an arbitrary element y ∈ H . Then we
obtain that
σ(xy) = (σxL) (y) =
(
σ(x)Lσ
π(x)
)
(y) = σ(x)σπ(x)(y).
Thus σ is a skew-morphism of H .
Vice-versa, if σ is a skew-morphism then the equation (2.1) holds proving G = 〈σ〉HL
= HL〈σ〉. It follows thatHL ≤ 〈σ〉HL ≤ Sym(HL). Hence,G = 〈σ〉HL is an overgroup
of HL with the point stabilizer Ge = 〈σ〉.
Following [4], we shall refer to the group 〈HL, σ〉 in (ii) as the skew product group
induced by σ.
Corollary 2.3. Let σ be a skew-morphism of a groupH , and T be an orbit of 〈σ〉 such that
〈T 〉 = H . Then 〈σ〉 acts faithfully on T . In particular, ord(σ) = |T |.
Remark. In the case where H is an abelian group we will consider the right action of
permutations of H , that is, the product π1π2 of two permutations evaluated from the left
to the right; in this case we write xπi (or ocasionally xπi ) for the image πi(x), x ∈ H ,
and we have x (π1π2) = (xπ1)π2. In this context we obtain the right permutation group
〈HR, σ〉 acting on H , where HR is the right representation of H consisting of permu-
tations hR (that is xhR = xh, x ∈ H) and where σ acts on the right (that is xσ =
σ(x), x ∈ H). Since H is abelian, we have HL = HR, and from (i) in Proposition 2.1
we get y (xR σk) = σk(yx) = σk(xy) = σk(x)σ
∑k−1
i=0 π(σ
i(x))(y) for all x, y ∈ H . The
latter equals y (σ
∑k−1
i=0 π(σ
i(x)) σk(x)R), hence giving us the commuting rule in the right
permutation group 〈HR, σ〉,
xR σ
k = σ
∑k−1
i=0 π(σ
i(x)) σk(x)R for all x ∈ H. (2.2)
Conversely, if G is a right permutation group of H such that HR ≤ G and the stabilizer Ge
is a cyclic group, then any generator of Ge is a skew-morphism of H .
LetG ≤ Sym(H) be a transitive permutation group, and let B be an imprimitivity block
system (or block system for short) of G. Then denote by GB the corresponding kernel, that
is, GB = {g ∈ G | g(B) = B for all B ∈ B}. Furthermore, denote the permutation of B
induced by the action of g ∈ G on B by gB, and set GB = {gB | g ∈ G}. For a block B,
denote by G{B} the set-wise stabilizer of B in G, and for a subgroup H ≤ G{B}, denote
the restriction of H to B by HB . Note that in the case where HL ≤ G ≤ Sym(H) and H
is abelian, any block system B is formed by the K-cosets of a subgroup K ≤ H . In this
case we identify B by the factor groupH/K, and it follows readily that (HL)B = (H/K)L
and (H/K)L ≤ GB ≤ Sym(H/K).
Skew-morphisms of abelian groups satisfy the following properties (see also [3]).
Proposition 2.4. Let σ be a skew-morphism of an abelian group H with power function
π, and let B be a block system of the skew product group G = 〈HL, σ〉 formed by the
K-cosets of a subgroup K ≤ H . Then the following hold.
(i) The restriction σK is a skew-morphism of K. If there is a block system of G formed
by the K ′-cosets of a subgroup K ′ ≤ H such that H = K × K ′, then σK is an
automorphism of K if and only if KL / G.
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 333
(ii) The permutation σB is a skew-morphism of H/K; moreover, its power function π
satisfies π(Kx) ≡ π(x) (mod ord(σB)) for all x ∈ H.
Proof. (i): We have σK is an automorphism ofK if and only if π(x) ≡ 1 (mod ord(σK))
for all x ∈ K. If KL / G, then σxLσ−1 = σ(x)L ∈ KL for all x ∈ K and by the
previous remark, σK is an automorphism of K. Suppose next that σK is an automorphism
of K. Then G is a subgroup of GK{K} × G
K′
{K′}. As G
K
{K} = 〈σ
K ,KKL 〉, it follows that
KKL / G
K
{K}, hence KL / G.
(ii): GB is an overgroup of (H/K)L in Sym(H/K). The group GB has cyclic sta-
bilizer 〈σB〉, giving that σB is a skew-morphism, see Proposition 2.2. Since σ is a skew-
morphism, we have σxL = σ(x)Lσπ(x) in G. This implies in GB that σB (Kx)L =
σB(Kx)L (σ
B)π(x). However σB (Kx)L = σB(Kx)L (σB)π(Kx), so the result fol-
lows.
2. Schur rings
Let H be a group written in multiplicative notation and denote the identity element
by e. Denote by Z(H) the group ring of H over the ring Z. The group ring Z(H) is
also a Z-module with scalar multiplication a(
∑
h∈H ahh) =
∑
h∈H(aah)h, a ∈ Z. The
Z-submodule of Z(H) spanned by the elements η1, . . . , ηr ∈ Z(H) will be denoted by
〈η1, . . . , ηr〉. Given a subset S ⊆ H , we write S for the Z(H)-element
∑
h∈H ahh defined
by ah = 1 if h ∈ S, and ah = 0 otherwise. Such elements in Z(H) will be called simple
quantities.
By a Schur ring (for short S-ring) A over H of rank r we mean a subring of the group
ring Z(H) such that there exist subsets T1, . . . , Tr of H satisfying the following axioms:
1. A has a Z-module basis of simple quantities: A = 〈T1, . . . , Tr〉.
2. T1 = {e}, and
∑r
i=1 Ti = H .
3. For every i ∈ {1, . . . , r} there exists j ∈ {1, . . . , r} such that T−1i = {x−1 | x ∈
Ti} = Tj .
The subsets T1, . . . , Tr are the basic sets of A, and we use the notation Bsets(A) ={
T1, . . . , Tr
}
. A subgroup K ≤ H is an A-subgroup if K ∈ A. In this case AK =
A ∩ Z(K) is an S-ring over K, also called an induced S-subring of A.
It is trivial that the whole group ring Z(H) and the Z-submodule 〈e,H \ {e}〉 are S-
rings over H . The latter one is the trivial S-ring over H . Further examples can be obtained
from an overgroup G of HL in Sym(H) as follows. Let Ge denote the stabilizer of e in
G, and let T1 = {e}, T2, . . . , Tr be the orbits of Ge. Then due to a result of Schur, the Z-
submodule 〈T1, . . . , Tr〉 is an S-ring over H (see [17, Theorem 24.1]). This is also called
the transitivity module over H induced by Ge, denoted by V (H,Ge). Thus
V (H,Ge) = 〈T | T ∈ Orb(Ge) 〉.
It is interesting to note that not all S-rings over H arise in this way, and an S-ring A is
called Schurian ifA = V (H,Ge) for some overgroup G of HL in Sym(H). Note that, if a
subgroup K ≤ H satisfies K ∈ V (H,Ge), then K is a block of the permutation group G.
334 Ars Math. Contemp. 4 (2011) 329–349
We conclude this section with two structure theorems of Evdokimov and Ponomarenko
about S-rings over cyclic groups. This requires some more general notation. Let A be an
S-ring of a group H , and let E and F be two A-subgroups of H such that E / H , F / H ,
E ∩ F = 1, and EF = H . The S-ring A is the tensor product of the induced S-subrings
AE and AF , denoted by A = AE ⊗ AF , if for any basic set Ti of A we have Ti = TjTk
for some basic sets Tj ⊂ E and Tk ⊂ F .
For an arbitrary subset X ⊆ H its radical is defined as rad(X) = {y ∈ H : Xy =
yX = X}. Equivalently, radX is the largest subgroup K of H such that X is the union of
both right and left cosets of K.
An S-ring A of a group H is said to satisfy the U/L-condition ([11, Definition 5.2]) if
the following hold:
1. L ≤ U ≤ H , and L / H ,
2. L and U are A-subgroups,
3. L ≤ rad(Ti) for any basic set Ti ∈ Bsets(A) such that Ti ⊆ H \ U .
If, moreover, L 6= 1, U 6= H , then we say A satisfies the U/L-condition nontrivially. In
[16], the S-ring A satisfying the above conditions was introduced as the wedge product
of S-rings AU and the quotient S-ring A/L. The latter S-ring is defined over the factor
group H/L by having basic sets Ti/L = {Lt | t ∈ Ti}, i ∈ {1, . . . , r} (see [15, Proposi-
tion 3.5]). If in addition U = L, then we say that A is the wreath product of the quotient
S-ring A/L with the induced S-subring AL, and shall write A = A/L o AL.
Now, suppose that H is an abelian group and let Ti be a basic set of an arbitrary S-ring
A over H . Then it follows for any number m coprime with |H| that (see [17, Theorem
23.9 (a)]),
T
(m)
i := {x
m | x ∈ Ti } is a basic set of A. (2.3)
Let H = Zn. Then (2.3) implies that rad(Ti) is the same for all basic sets Ti of A that
contain a generator of Zn. This radical is the radical of A and denoted by radA.
The first theorem describes S-rings with trivial radical (see [11, Corollary 6.4]).
Theorem 2.5. LetA be an S-ring over Zn. Then radA = 1 if and only if Zn is decomposed
into a direct product Zn = E1×· · ·×Ek ofA-subgroupsE1, . . . , Ek, andA is decomposed
into the tensor productA = AE1⊗· · ·⊗AEk of induced S-subringsAE1 , . . . ,AEk , where
AE1 is normal with trivial radical, and AEi is trivial for all i ∈ {2, . . . , k}.
In general, an S-ring A over H is normal if HL E Aut(A), where Aut(A) is the
subgroup in Sym(H) formed by all permutations preserving the relations (see [13])
Ri =
{
(h, ht) | h ∈ H, t ∈ Ti
}
, Ti ∈ Bsets(A).
Note that if A = V (H,Ge) for some HL ≤ G ≤ Sym(H), then G ≤ Aut(A).
For the case of nontrivial radical the following statement holds true (see [11, Corol-
lary 5.5]).
Theorem 2.6. An S-ring A over Zn satisfies some U/L-condition nontrivially if and only
if radA > 1.
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 335
3 S-rings and skew-morphisms of Zn
Let σ be a skew-morphism of a group H . Then the skew product group G = 〈HL, σ〉 is
an overgroup of HL in Sym(H) with stabilizer Ge = 〈σ〉. Thus σ induces the transitivity
module V (H, 〈σ〉), which we call the S-ring generated by σ, and for short denote it also
by V (σ). In this section we study these S-rings in the case when H is a cyclic group.
First we set some notation. For the rest of the paper the cyclic group of order n is the
additive group Zn, written also as Zn = {0, 1, . . . , n − 1}, and will also denote the ring
of integers modulo n. As usual, we denote by Z∗n the multiplicative group of invertible
elements in Zn. For x ∈ Zn we write ord(x) for the order of x as an element in Zn, and
ord∗(x) for the order of x as an element in Z∗n whenever x ∈ Z∗n. For a ∈ Zn we write αa
for the homomorphism from the cyclic group Zn onto the cyclic group Zn/(n,a) defined by
αa(x) = ax, x ∈ Zn.
The mapping αa is an automorphism if and only if a ∈ Z∗n. For a divisor d of n let Cd
denote the unique additive subgroup of Zn of order d. Thus we have Cd =
{
x(n/d) |
x ∈ {0, . . . , d− 1}
}
. Let τ be the permutation in Sym(Zn) acting as follows
τ(x) = x+ 1, x ∈ Zn.
By Proposition 2.2 a skew-morphism σ of Zn with a power function π induces the skew
product group G = 〈τ, σ〉.
By (i) in Proposition 2.1, for a fixed x ∈ Zm, m = ord(σ), and y ∈ Zn we have
σx(y + z) = σx(y)σπ(y)+π(σ(y))+···+π(σ
x−1(y))(z) for all z ∈ Zn.
Thus the following commuting rule holds in G,
σxτy = τσ
x(y)σπ(y)+π(σ(y))+···+π(σ
x−1(y)) for all x ∈ Zm, y ∈ Zn, (3.1)
where m = ord(σ). In particular, the characteristic identity of a skew-morphism σ of Zn
can be expressed as
στy = τσ(y)σπ(y) for all y ∈ Zn.
Suppose that B is a block system of G = 〈τ, σ〉, and B is formed by the Cd -cosets
for some subgroup Cd ≤ Zn. Then we write σ|d for the skew-morphism σCd , see (i)
in Proposition 2.4. Also, we write σ|n/d for the skew-morphism σB of the factor group
Zn/Cd ∼= Zn/d, see (ii) in Proposition 2.4.
In our first result we describe skew-morphisms inducing S-rings with trivial radical.
Proposition 3.1. Let σ be a skew-morphism of Zn with radV (σ) = 1. Then σ is an
automorphism of Zn .
Proof. We set G = 〈τ, σ〉 and A = V (σ). Assume first that radV (σ) = 1. By The-
orem 2.5, the group Zn decomposes into a direct product Zn = Cn1 × · · · × Cnk of
A-subgroups Cn1 , . . . , Cnk so that
A = ACn1 ⊗ · · · ⊗ ACnk ,
336 Ars Math. Contemp. 4 (2011) 329–349
where the induced S-subring ACn1 is normal, and ACni is trivial for all i ∈ {2, . . . , k}.
It is obvious that ACni = V (σ|ni) for all i ∈ {1, . . . , k}. In particular, the skew product
group Gi = 〈(Cni)L, σ|ni〉 ≤ Aut(ACni ) for all i ∈ {1, . . . , k}.
For i = 1 the subring ACn1 is normal, hence (Cn1)L E Aut(ACn1 ). Therefore
(Cn1)L E 〈(Cn1)L, σ|n1〉, so σ|n1 is an automorphism of Cn1 . By (i) in Proposition 2.4,
(Cn1)L / G.
Let i > 1. Then ACni = V (σ|ni) is trivial. Thus the subring ACni is generated by 0
and Cni \ {0}, and so Cni \{0} is an orbit of σ|ni . It follows that ord(σ|ni) = |Cni \{0}|,
and we obtain that Gi is a sharply 2-transitive Frobenius group. Therefore Gi must have a
regular elementary abelian normal subgroup, say Ei. Furthermore, |Gi| = |Ei|(|Ei| − 1),
hence Ei is a Sylow p-subgroup of Gi. We conclude that ni = |Ei|, (Cni)L = Ei / Gi.
Hence σ|ni is an automorphism of Cni , and (Cni)L / G by (i) in Proposition 2.4.
To summarize, (Cni)L / G for all i ∈ {1, . . . , k}. Therefore, (Zn)L / G, equivalently,
σ is an automorphism of Zn.
It follows from Theorem 2.6 that a generating orbit of a pure skew-morphism σ of Zn
is a union of cosets of a nontrivial subgroup L ≤ Zn.
We give next three corollaries which seem to be of independent interest.
Corollary 3.2. All skew-morphisms of Zn have power functions with nontrivial kernel.
Proof. Let σ be a skew-morphism of Zn with power function π. To simplify notation we
put H = Zn. The skew product group G = 〈HL, σ〉 = HL 〈σ〉. Let m = ord(σ). By
Corollary 2.3, m < n. Consider G acting on the right cosets of HL in G. Denote by
G̃ the induced permutation group. Then G̃ is of degree m, in particular, 〈̃σ〉 is a regular
cyclic subgroup. The point stabilizer of the coset HL in G̃ is H̃L, hence it is cyclic. Also,
H̃L ∼= HL/ coreG(HL), where coreG(HL) =
⋂
g∈G g
−1HLg, or in other words, the
largest normal subgroup of G contained in HL. Any generator of H̃L becomes a skew-
morphism of 〈̃σ〉, hence |H̃L| < m < n = |HL|. Therefore, there exists a nontrivial
subgroup K ≤ H = Zn such that KL / G. It is obvious that K ≤ kerπ (see the proof of
(i) in Proposition 2.4).
The next fact was important in the study of regular cyclic maps (see [2]).
Corollary 3.3. Let σ be a skew-morphism σ of Zn, and T be an orbit of σ such that
〈T 〉 = Zn. Then there exists t ∈ T such that 〈t〉 = Zn.
Proof. We prove the statement by induction on n. The statement is trivially true if n is a
prime. Thus we assume that n is a composite number, and that the statement is true for all
groups Zn′ such that n′ < n.
If radV (σ) = 1 then by Proposition 3.1 σ is an automorphism of Zn, and we are
done. Let radV (σ) 6= 1. Then V (σ) satisfies the Cu/Cl-condition for subgroups 1 <
Cl ≤ Cu < Zn. Since 〈T 〉 generates Zn, the set T/Cl = {x + Cl| x ∈ T} forms a
generating orbit of Zn/Cl in the quotient skew-morphism. By the induction hypothesis
there exists t ∈ T such that 〈t + Cl〉 = Zn/Cl. If 〈Cl + t〉 < Zn, then 〈Cl + t〉/Cl <
Zn/Cl, and Cl + t is a nongenerating element of Zn/Cl, a contradiction. It follows that
Zn = 〈t + Cl〉 = 〈t〉 + Cl. Let m = ord(t). Then n = lm/(l,m). Set t′ = nl + t. Then
ord(t′) = lm/(l,m) = n, so 〈t′〉 = Zn. Now, by the Cu/Cl condition we find Cl+t ⊆ T,
hence t′ ∈ T , proving the result.
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 337
Corollary 3.4. Let σ be a skew-morphism of Zn. Then ord(σ) is a divisor of nφ(n).
Moreover, if (ord(σ), n) = 1, then σ is an automorphism of Zn.
Proof. We prove the statement by induction on n. If n is a prime, then by Corollary 3.2
kerπ = Zn, hence σ is an automorphism of Zn and ord(σ) | φ(n). Thus we assume that
n is a composite number, and that the statement is true for all groups Zn′ such that n′ < n.
Let σ be an arbitrary skew-morphism of Zn with power function π. In the case where
radV (σ) = 1, by Proposition 3.1 the skew-morphism σ is an automorphism of Zn. It
follows that ord(σ) | φ(n).
Let radV (σ) 6= 1. Then V (σ) satisfies the Cu/Cl-condition for subgroups 1 < Cl ≤
Cu < Zn. Consider the skew-morphism σ|n/l of the factor group Zn/Cl ∼= Zn/l. Let
T be an orbit of σ such that 〈T 〉 = Zn. By Corollary 2.3, ord(σ) = |T |. Then T/Cl is
an orbit of σ|n/l such that 〈T/Cl〉 = Zn/Cl, and hence ord(σ|n/l) = |T/Cl|. As V (σ)
satisfies the Cu/Cl-condition, |T/Cl| = |T |/l, and we obtain that ord(σ|n/l) = ord(σ)/l.
Induction gives
ord(σ)/l = ord(σ|n/l) | (n/l)φ(n/l).
This implies that ord(σ) | nφ(n/l), hence ord(σ) | nφ(n).
As noted after Proposition 3.1, if σ is a pure skew-morphism of Zn, then a generating
orbit T of σ must be a union of L-cosets for a subgroup 1 < L ≤ Zn. By Corollary 2.3,
ord(σ) = |T | is divisible by |L|, in particular |L| | (ord(σ), n). Now, it is clear that if
(ord(σ), n) = 1, then σ must be an automorphism of Zn.
If n is a square-free integer and σ is a skew-morphism of Zn, then the fact that σ is an
automorphism of Zn can be read off quickly from the generated S-ring V (σ).
Proposition 3.5. Let n be a square-free number and σ be a skew-morphism of Zn. Then σ
is an automorphism of Zn if and only if Cd ∈ V (σ) for all subgroups Cd ≤ Zn.
Proof. The implication⇒ is obvious.
The converse implication⇐ follows by an easy induction on the order n. We get that
all subgroups of (Zn)L of prime index are normal in G = 〈τ, σ〉. Since n is square-free,
this implies (Zn)L E G, equivalently, σ is an automorphism of Zn.
There is yet another S-ring which can be associated with a skew-morphism σ of an
arbitrary group H . Consider the skew product group G = 〈HL, σ〉 = HL〈σ〉. The group
G has a well-known right action on the set G/HL of right HL-cosets in G as for x, y ∈ G,
(HLx)
y = HLxy. As G = HL 〈σ〉 and |HL ∩ 〈σ〉| = 1, the elements σi form a complete
set of coset representatives of HL in G. Thus the set G/HL can be identified with 〈σ〉, and
the group G admits a corresponding right action on 〈σ〉. (We remark that the only property
we are using here is that |HL ∩ Ge| = 1, hence the same right action can be defined on
arbitrary point stabilizer Ge, and Ge need not be generated by one element.) Formally, for
σi ∈ 〈σ〉 and g ∈ G we set (σi)g = σj , where the integer j satisfies
HL(σ
i)g = HL σ
ig = HLσ
j .
For g ∈ G, denote by g̃ the permutation of 〈σ〉 induced by g. The subgroup 〈σ〉 ≤
G is faithful, and 〈̃σ〉 = 〈σ〉R. The group HL is not always faithful, in fact H̃L ∼=
338 Ars Math. Contemp. 4 (2011) 329–349
HL/ coreG(HL), where coreG(HL) =
⋂
g∈G g
−1HLg, or in other words, the largest nor-
mal subgroup of G contained in HL. The stabilizer of id ∈ 〈σ〉 equals H̃L, hence we
obtain the S-ring V (〈σ〉, H̃L) over 〈σ〉. Notice that coreG(HL) ≤ KL, where K = kerπ,
π is the power function of σ, and that the conjugate subgroup σKLσ−1 ≤ HL. If H ∼= Zn
is a cyclic group, then every subgroup of HL is uniquely determined by its order, in par-
ticular, |σKLσ−1| = |KL| yields that σKLσ−1 = KL. This implies, in turn, that KL is
a normal subgroup in G, KL ≤ coreG(HL), and so coreG(HL) = kerπ. Then H̃L is
cyclic, and thus any of its generators is a skew-morphism of 〈σ〉 (see the remark following
Corollary 2.3). Recall that HL = 〈τ〉, where τ is the permutation x 7→ x+ 1, x ∈ Zn. Let
ord(σ) = m, and let ψ be the isomorphism from 〈σ〉 to Zm sending σx to x ∈ Zm. Then
the function
σ∗ = ψ
−1 τ̃ ψ
is a skew-morphism of Zm. We shall call σ∗ the skew-morphism derived from σ. The
skew-morphism σ is related to its derived skew-morphism σ∗ as follows.
Proposition 3.6. Let σ be a skew-morphism of Zn with ord(σ) = m, and π be the power
function of σ. Then the derived skew-morphism σ∗ of Zm and its power function π∗ are
given by
(σ∗)
y(x) = π(y) + π(σ(y)) + · · ·+ π(σx−1(y)), (3.2)
π∗(x) ≡ σx(1) (mod n/| kerπ|). (3.3)
Proof. Using commuting rule (3.1) in the skew product groupG = 〈τ, σ〉 we conclude that
〈τ〉σxτy = 〈τ〉σπ(y)+π(σ(y))+···+π(σ
x−1(y)).
This shows that τ̃ y : σx 7→ σπ(y)+π(σ(y))+···+π(σx−1(y)), and (3.2) follows.
Notice that 〈σ̃, τ̃〉 = 〈〈σ〉R, τ̃〉 is a right permutation group with skew-morphism τ̃ .
Thus by the commuting rule (2.2),
σ̃ x τ̃ = τ̃ π∗(x) σ̃ σ∗(x) for all x ∈ Zm.
Combining this with (3.1) gives us π∗(x) ≡ σx(1) (mod ord(τ̃)). Now, ord(τ̃) =
[(Zn)L : coreG((Zn)L)], and (3.3) follows as coreG((Zn)L) ∼= kerπ.
Note that the derived skew-morphism σ∗ is the identity if and only if σ is an automor-
phism.
4 S-rings V (σ) over Zpe
In this section our goal is to prove the following theorem about S-rings V (σ) where σ ∈
Skew(Zpe), p is an odd prime.
Theorem 4.1. If σ ∈ Skew(Zpe), where p is an odd prime, then V (σ) = V (α) for some
α ∈ Aut(Zpe).
Theorem 4.1 suggests the following definition.
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 339
Definition 4.2. A permutation σ of a finite group H is a near automorphism if σ is a skew-
morphism ofH, and the orbits of σ are equal to the orbits of an automorphism α ∈ Aut(H)
(or equivalently, V (σ) = V (α) holds).
In this context Theorem 4.1 can be rephrased as every skew-morphism of Zpe , p is an odd
prime, is a near automorphism.
We first prove several properties of skew-morphisms of Zpe , p is an odd prime.
Lemma 4.3. Let σ ∈ Skew(Zpe) be a skew-morphism with power function π. Then all
subgroups Cpi , i ∈ {0, 1, . . . , e} are blocks of 〈τ, σ〉.
Proof. We prove the lemma by induction on the exponent e. The case e = 1 is trivially
true. By Corollary 3.2, the power function π of σ has nontrivial kernel, say K. Therefore
K is normal in the group G = 〈τ, σ〉, in particular, Cp is a block of G. Now, applying
induction to σ|pe−1 if e > 1 yields the result.
In the next lemma we consider skew-morphisms of p-power order.
Lemma 4.4. Let σ ∈ Skew(Zpe) of order pf with power function π. Then the following
hold:
(i) The orbit T generated by 1 is the coset T = 1 + Cpf , in particular σi(1) ≡ 1
(mod pe−f ) for i = 0, 1, . . . , pf − 1;
(ii) For all x ∈ Zpe , π(x) ≡ 1 (mod p); in particular, if the order of σ is p then σ is an
automorphism;
(iii) For every i ∈ N, σi is a skew-morphism of Zpe , moreover ord(σi(1)) = pf/(pf , i).
Proof. We put G = 〈τ, σ〉. Observe that f < e.
(i): By Corollary 2.3, |T | = ord(σ) = pf . By Lemma 4.3 Cpe−1 = 〈p〉 is fixed by σ
and hence T ⊆ Z∗pe . Let c be a generator of Z∗pe . By (2.3) T, cT, c2T, . . . , cp
e−1(p−1)−1T
are basic sets in V (σ) and the union of these sets forms Z∗pe . It follows that the least
power of c fixing T is d = c(p−1)p
e−f−1
of multiplicative order pf . In particular T =
{1, d, d2, . . . , dpf−1}. Since Z∗pe is cyclic, it contains a unique subgroup of order pf ,
namely 1 + Cpf . It follows that T = 1 + Cpf .
(ii): Consider the skew-morphism σ∗ derived from σ. Then σ∗ is a skew-morphism of
Zpf of p-power order. Because of (3.2), σ
y
∗(1) = π(y) for all y ∈ Zpe . Now, the result
follows by (i).
(iii): If i is coprime to p then σi is a generator of the stabilizer of 0 in G and the result
follows by Proposition 2.2. For i = p` for some ` > 0 by (i) in Proposition 2.1 σi is a
skew-morphism if and only if p` divides
∑p`−1
j=0 π(σ
j(x)) which holds true by (ii). Clearly,
the orbit of σp
`
generated by 1 consists of the elements of the multiplicative subgroup of
Z∗pe generated by dp
`
, and we are done.
Next we determine the skew-morphisms σ ∈ Skew(Zpe) of order p2. Denote by
Fun(Zn,Zn) the set of all functions from Zn to Zn. For f, g ∈ Fun(Zn,Zn), let fg and
f + g be the functions in Fun(Zn,Zn) defined as (fg)(x) = f(g(x)) and (f + g)(x) =
340 Ars Math. Contemp. 4 (2011) 329–349
f(x) + g(x), respectively, for all x ∈ Zn. Further, if c ∈ Zn, then let cf be the function in
Fun(Zn,Zn) defined as (cf)(x) = cf(x) for all x ∈ Zn.
Let f ∈ Fun(Zn,Zn). We introduce the sum operator ∇ as the function in Fun(Zn,
Zn) defined as
(∇f)(0) = 0, and
(∇f)(x) = f(0) + f(1) + · · ·+ f(x− 1) if x ∈ Zn, x 6= 0.
Note that ∇ satisfies∇(f + g) = ∇f +∇g for all f, g ∈ Fun(Zn,Zn).
Let us put n = pe for the rest of the section, where p is an odd prime. For a ∈ Z∗pe and
b ∈ Zpe , b ∈ Cp define the function
σa,b = αa +∇αb.
Let Ge =
{
σa,b | a ∈ Z∗pe , ord
∗(a) = pi, i ∈ {0, 1, . . . , e− 1}, b ∈ Cp
}
.
Lemma 4.5. With the above notation Ge is a subgroup of Sym(Zpe),Ge ∼= Zpe−1 × Zp
with the composition rule σa,bσc,d = σac,b+d. In particular, ord(σa,b) = ord
∗(a) if a 6= 1
and ord(σa,b) = ord(b) if a = 1.
Proof. Clearly, the identity permutation id = σ1,0, hence id ∈ Ge. Now, it is enough to
prove that σa1,b1σa2,b2 = σa1a2,b1+b2 for all σa1,b1 , σa2,b2 ∈ G. Since b ∈ Cp, we have
(∇αb)(x) = (∇αb)(x′) if x ≡ x′ (mod p).
Since a2 ∈ 1 + Cpi for some i ∈ {0, 1, . . . , e − 1} we have σa2,b2 ≡ id (mod p). It
follows that (∇αb1)σa2,b2 = ∇αb1 . Thus
σa1,b1σa2,b2 = (αa1 +∇αb1)σa2,b2 = αa1σa2,b2 +∇αb1 .
Since a1b2 = b2 in Zpe ,
αa1σa2,b2 = αa1a2 +∇αa1b2 = αa1a2 +∇αb2 .
These yield σa1,b1σa2,b2 = αa1a2 +∇αb2 +∇αb1 = αa1a2 +∇αb1+b2 . It follows that Ge
is a group generated by the automorphisms σa,0 and elements σ1,b.
Lemma 4.6. The permutations σa,b in Ge are pure skew-morphisms of Zpe if and only if
a ∈ 1 + Cpf , where f > 1 and b 6= 0.
Proof. Let b 6= 0 and a ∈ 1 + Cpf , where f > 1. To simplify notation we set σ = σa,b.
Let a′ = b + 1. Notice that a′ ∈ Z∗pe and ord
∗(a′) = p. By Lemma 4.5, σp = αap +
∇αpb = αap . Thus σ` = αa′ for a suitable ` ∈ N. Since the order of σ is pf for some
f ∈ {2, . . . , e− 1}, we get that pf−1 divides `.
It follows that (xb + 1) = a′x for all x ∈ Zpe . We want to show that σ(x + 1) =
a(1 + a′ + a′ 2 + · · ·+ a′ x) for all x ∈ Zpe . We have
a(1 + a′ + a′ 2 + · · ·+ a′ x) = a(1 + (b+ 1) + (b+ 1) 2 + · · ·+ (b+ 1) x)
= a(x+ 1) + a(b+ 2b+ 3b+ · · ·+ xb)
= a(x+ 1) + ab(1 + 2 + 3 + · · ·+ x)
= a(x+ 1) + b(1 + 2 + 3 + · · ·+ x)
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 341
If x 6= pe − 1 the last term is equal σ(x + 1) by definition. For x = pe − 1 we get
a((pe − 1) + 1) + b(1 + 2 + 3 + · · ·+ (pe − 1)) = 0 = σ(0).
Now we are ready to prove the commuting rule στ = τaσ`+1. This is equivalent to the
identity σ(x + 1) = a + a′σ(x). Inserting σ(x + 1) = a(1 + a′ + a′ 2 + · · · + a′ x) and
σ(x) = a(1 + a′ + a′ 2 + · · ·+ a′ x−1) we get the result.
Taking this to the ith power, i ∈ N we find σ〈τ〉 ⊆ 〈τ〉〈σ〉. A simple induction
argument gives σi〈τ〉 ⊆ 〈τ〉〈σ〉 for all i ∈ N. We conclude that 〈σ〉〈τ〉 = 〈τ〉〈σ〉, hence
σ is a skew-morphism of Zpe . Furthermore, σ takes 1 7→ a and 2 7→ 2a + b. Therefore, it
cannot be an automorphism for b 6= 0.
Clearly σa,0 is an automorphism. It remains to show that σa,b, where b 6= 0 and a ∈
1+Cp is not a pure skew-morphism. By Lemma 4.5 the order of σ is p. By Lemma 4.4(iii)
σa,b is an automorphism, a contradiction.
Lemma 4.7. The skew-morphisms of Zpe of order p2 are exactly the elements σa,b in Ge of
order p2.
Proof. Let σ(1) = a. By (ii) in Lemma 4.4, a ∈ Z∗pe with ord
∗(a) = p2. Let σ′ =
σπ(1)−1. Because of (iii) in Lemma 4.4 we have σ′ is a skew-morphism of Zpe of order
at most p. Therefore σ′ is an automorphism of Zpe , and we may write σ′ = αa′ for some
a′ ∈ Z∗pe , ord
∗(a′) = pi, i ≤ 1.
By (3.1), σ satisfies στ = τσ(1)σπ(1). Therefore,
στ = τaσπ(1)−1σ = τaαa′σ = τ
a(αa′ + σ) = σ + κa + αb, (4.1)
where b = a′ − 1, and κa is the constant function κa(x) = a for all x ∈ Zpe . Notice
that b ∈ Cp, and hence we have αa′σ = αa′ as σ(x) ≡ x (mod p) for all x ∈ Zpe . It is
not difficult to see that the functional equation in (4.1) with σ as variable and initial value
σ(0) = 0 has a unique solution, and this is
σ = ∇(κa + αb) = ∇κa +∇αb = αa +∇αb.
We obtain that σ ∈ Ge and this completes the proof.
Proof of Theorem 4.1. We put G = 〈τ, σ〉, and let P be the Sylow-p-subgroup of G. By
Corollary 3.4, ord(σ) = pfd, where d | (p − 1). The group P contains (Zpe)L and has
stabilizer P0 = 〈σd〉. It follows from Proposition 2.2 that σd is a skew-morphism of Zpe of
order pf . If f = 0, then σ is an automorphism of Zpe , see Corollary 3.4. Thus we assume
that f ≥ 1.
We prove the theorem by induction on e. If e = 1, then σ is an automorphism of Zp,
and the proposition is trivially true. Let e > 1 and assume that the proposition holds for
any group Zpe′ for which e′ < e. We put A = V (σ). By Lemma 4.3, all subgroups Cpi
are blocks of G, equivalently, Cpi ∈ A for all i ∈ {0, 1, . . . , e}.
The further argument is divided into four steps.
(a) ACpe−1 = V (β) for some automorphism β in Aut(Cpe−1).
We have ACpe−1 = V (Cpe−1 , σ| pe−1). Thus (a) follows by induction.
342 Ars Math. Contemp. 4 (2011) 329–349
(b) A/Cp = V (γ) for some automorphism γ in Aut(Zpe/Cp).
We have A/Cp = V (Zpe/Cp, σ| p
e−1
). Thus (b) follows by induction.
(c) There exists a unique automorphism α in Aut(Zpe) such that α|pe−1 = β and α|p
e−1
=
γ.
First, there exists a unique automorphism α in Aut(Zpe) such that
p | ord(α) and α|p
e−1
= γ.
Then
β|p
e−2
=
(
σ|pe−1
)
|p
e−2
=
(
σ|p
e−1)
|pe−2 = γ|pe−2 .
The latter can be written as (
α|p
e−1)
|pe−2 =
(
α|pe−1
)
|p
e−2
.
Thus we have β|pe−2 =
(
α|pe−1
)
|pe−2 . This forces β = α|pe−1 if p2 | ord(α|pe−1), or
(α|pe−1 , pe−2) = (β, pe−2).
Assume that f ≥ 3. Then p3 | ord(σ), hence p3 | ord(α), and we are done as
p2 | ord(α|pe−1).
Let f = 2. Then (α|pe−1 , pe−2) = p. But σd is a skew-morphism of Zpe of order p2.
Applying Lemma 4.7 we find that ord(σ|pe−1 , pe−2) = p also.
Let f = 1. Then (α|pe−1 , pe−2) = 1. But σd is a skew-morphism of Zpe of order
p. Thus σd is an automorphism of Zpe , and we find that ord(σ|pe−1 , pe−2) = 1. This
completes (c).
(d) A = V (α).
This follows directly from (c) and the fact that every basic set T ∈ Bsets(A) where
〈T 〉 = Zpe is a union of Cp-cosets.
Remark. Theorem 4.1 does not hold for powers of 2. Take the permutation σ ∈ Sym(Z2e )
defined as
σ(x) =
{
x+ 2 if x /∈ C2e−1 ,
x if x ∈ C2e−1 .
It is easy to check that σ is a skew-morphism of Z2e , and its power function π satisfies
π(x) = 1 for all x ∈ C2e−1 , and π(x) = −1 for all x /∈ C2e−1 . Clearly, if e ≥ 3, then
V (σ) 6= V (α) for all α ∈ Aut(Z2e).
As a small application of Theorem 4.1 we determine the whole set Skew(Zp2). We
need to consider the automorphisms of the group
Me =
〈
x, y | xp
e
= yp = 1, xy = xp
e−1+1
〉
.
The next lemma is a special case of [1, Lemma 2.1].
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 343
Lemma 4.8. The automorphisms in Aut(Me) are given as the permutations θa,b,c, a ∈
Z∗pe , b ∈ Cp and c ∈ Zp defined as
θa,b,c(x
iyj) = xai+bj yci+j for all i ∈ Zpe , j ∈ Zp.
Therefore, |Aut(Me)| = (p− 1)pe+1.
Proposition 4.9. If p is an odd prime, then
Skew(Zp2) = Aut(Zp2)∪
{
αa+a∇αb | a ∈ Z∗p2 , ord
∗(a) = pd, d > 1, b ∈ Cp, b 6= 0
}
.
Proof. Let σ ∈ Skew(Zp2). We are going to show that σ is a permutation described in the
statement. We put G = 〈τ, σ〉 and let P be the Sylow p-subgroup of G. By Corollary 3.4,
ord(σ) | p(p−1). Let ord(σ) = pfd for a divisor d of p−1. Because of Sylow’s Theorem
P / G, and thus µ = σp acts on P by conjugation as an automorphism; denote by µ the
induced automorphism. If f = 0, then P = (Zp2)L, and σ = µ is an automorphism of Zp2
of order d.
Let f > 0. Then with a suitable choice σ0 ∈ 〈σd〉 we have
P =
〈
τ, σ0 | τp
2
= σ p0 = 1, τ
σ0 = τp+1
〉 ∼=M2.
By Lemma 4.8, µ = θu,v,w for some u ∈ Z∗p2 , v ∈ Cp and w ∈ Zp. Now, ord(µ) =
ord(µ) = d, and µ fixes σ0. These imply that u ∈ Z∗pe has ord
∗(u) = d, and v = 0. If
u = 1, then µ = θ1,0,1 is the identity, and in this case σ is an automorphism of Zp2 of order
p. Let u > 1. Then µ = θu,0,w for some w ∈ Zp. Then
µτµ−1 = µ(τ) = θu,0,w(τ) = τ
u αwp+1. (4.2)
Using the fact that µ(0) = 0, µ is determined by (4.2). We obtain that µ(x) = u(1 + (p+
1)w + · · ·+ (p+ 1)w(x−1)) for all x ∈ Zp2 , x 6= 0. Because (p+ 1)y = yp+ 1 in Zp2 for
all y ∈ Zp2 , we may write µ = αu + u∇αb, where b = wp is in Cp. This gives σ = αa′µ
for a′ ∈ Z∗p2 , ord
∗(a′) = p. Therefore σ = αa + a∇αb, where a = a′u. Now, σ is a pure
skew-morphism if and only if b 6= 0.
Conversely, let σ = αa + a∇αb, a ∈ Z∗p2 , ord
∗(a) = pd, d > 1, b ∈ Cp, b 6= 0. Then
a = a′a′′ for a′, a′′ ∈ Z∗p2 with ord(a′) = p and ord
∗(a′′) = d. From this σ = αa′σ′ for
σ′ = αa′′+a
′′∇αb. From the above discussion it follows that σ′ normalizesP = 〈τ, αp+1〉.
This implies 〈τ, σ〉 = 〈P, σ′〉 = P 〈σ′〉 = 〈τ〉〈σ〉, so that σ is a skew-morphism of Zp2 .
The formula for the number |Skew(Zp2)| of all skew-morphisms of Zp2 follows di-
rectly from Proposition 4.9.
Corollary 4.10. If p is an odd prime, then |Skew(Zp2)| = (p− 1)(p2 − 2p+ 2).
5 S-rings V (σ) over Zpq
We recall that the wreath product Z(Zq) o Z(Zp) of S-rings Z(Zq) and Z(Zp) is the S-ring
over Zpq that has basic sets {x}, x ∈ Cp and the cosets Cp+y, y ∈ Zpq \Cp. Similarly the
wreath product Z(Zp) o Z(Zq) is the S-ring over Zpq that has basic sets {x}, x ∈ Cq and
the cosets Cq + y, y ∈ Zpq \ Cq . Two distinct primes p, q will be called disjoint if neither
q | (p− 1), nor p | (q − 1).
344 Ars Math. Contemp. 4 (2011) 329–349
Theorem 5.1. Let q < p be distinct primes and σ be a skew-morphism of Zpq . Then
V (σ) = V (α) for some α ∈ Aut(Zpq), unless q | (p− 1) and V (σ) = Z(Zq) o Z(Zp).
Proof. Let σ be a pure skew-morphism of Zpq of order ord(σ) = m. By Proposition 3.5,
not all subgroups of Zpq are V (σ)-subgroups. The S-ring V (σ) is neither trivial, hence
either Cp ∈ V (σ) and Cq /∈ V (σ), or Cp /∈ V (σ) and Cq ∈ V (σ). We show that in the
first case q | (p−1) and V (σ) = Z(Zq) oZ(Zp) as described above. Then by the symmetry,
the second case implies p | (q − 1), a contradiction.
We put G = 〈τ, σ〉, and let B be the block system of G formed by the Cp-cosets. Thus
m = pd, for some d | (q − 1). Let GB denote the kernel of G acting on B. It is obvious
that τ q is in GB. By Proposition 2.4.(ii), σB is a skew-morphism of Zpq/CP ∼= Zq . It
follows from the classification of S-rings over Zpq that any basic set of V (σ), which is
not contained in Cp, is a union of Cp-cosets (e.g. [13]). Let T be an orbit of σ such that
〈T 〉 = Zpq . By Corollary 2.3, |σ| = |T | and |σB| = |T/Cp| = |T |/p. This implies that
|G/GB| = |G|/p2 and |〈σ〉 ∩GB| = |〈σ〉 : 〈σB〉| = p, and thus σd is in GB. We conclude
that
GB = 〈τ q, σd〉 = 〈τ q〉 × 〈σd〉 ∼= Zp × Zp.
Consider the group N = 〈τ〉GB. Then N = 〈τ〉〈σd〉, and σd is a pure skew-morphism
of Zpq of order p. Clearly, V (σd) = Z(Zq) o Z(Zp).
Since N = GB o 〈τp〉, the element τp acts on GB by conjugation as a group auto-
morphism. We may identify τp with a matrix A(τp) in GL(2, p) such that for all (i, j) ∈
Zp × Zp, (
τ qiσdj
) τp
= τ qi
′
σdj
′
if
(
i′
j′
)
= A(τp)
(
i
j
)
.
Now, τ q is fixed by τp and 〈σd〉 is not normalized by τp. Also, ord(A(τp)) =
ord(τp) = q. These conditions imply that
A(τp) =
(
1 a
0 b
)
, a, b ∈ Z∗p, bq = 1, b 6= 1.
It also follows that q | (p− 1).
Let A = A(τp). The group N = 〈τ q, σd〉 o 〈τp〉, and we may identify N with
the group of all pairs (x,Ai), where x = (x1, x2)> is in the vector space V = Z2p, and
i ∈ {0, 1, . . . , q − 1}. The group operation ∗ is defined as
(x,Ai) ∗ (y,Aj) = (x+Aiy,Ai+j).
Note that e1 = (1, 0) and e2 = (0, 1) in V correspond to τ q and σd, respectively. Denote
µ = σp.
Since N / G, G = N o 〈µ〉, and hence µ acts on N as a group automorphism. Denote
this automorphism by µ. Then µ satisfies the following conditions:
1. ord(µ) = d,
2. µ normalizes 〈(e1, I)〉,
3. µ fixes (e2, I),
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 345
To see 1. observe that if ord(µ) < d, then a subgroup 1 < M < 〈µ〉 centralizes N . In
particular, M centralizes τ , hence M is semiregular, which is impossible. Condition 2.
follows from the fact that 〈τ q〉 / G. Finally, Condition 3. follows from µσd = σdµ. The
above conditions imply
µ :
(x, I) 7→ (Bx, I), where B =
(
c 0
0 1
)
, for some c ∈ Z∗p.
(0, A) 7→ (v,Ak) for some v ∈ V, k ∈ {1, . . . , q − 1}
.
Since µ is an automorphism, it is defined by the above images, and we have
µ : (x,Ai) 7→ (Bx+ (I +Ak + · · ·+Ak(i−1))v,Aki).
In particular, (I +Ak + · · ·+Ak(q−1))v = 0. Then
µ
(
(x,A)
)
∗ µ
(
(y, I)
)
= µ
(
(x,A) ∗ (y, I)
)
= µ
(
(x+Ay,A)
)
.
Therefore (Bx + v,Ak) ∗ (By, I) = (Bx + AkBy + v,Ak) = (Bx + BAy + v,Ak).
From this BAB−1 = Ak, and thus
ac ≡ a(1 + b+ . . . , bk−1) (mod p),
bk ≡ b (mod p).
Now b 6= 1 and bq = 1 in Z∗p. As k < q, bk = b (mod p) implies k = 1. It follows that
c = 1. We have µ̄d : (0, A) 7→ (dv,A). Since µ̄d = id, dv = 0 and so v = 0. Thus
µ is the identity, and d = ord(µ) = 1. We obtain V (σ) = V (σd) = Z(Zq) o Z(Zp), as
claimed.
Let σ be a pure skew-morphism of Zpq . The order of σ equals |T |, where T is an orbit
of σ such that 〈T 〉 = Zpq (see Corollary 2.3). By Theorem 5.1 we obtain that a pure skew-
morphism of Zpq must have order either p or q. We conclude the section with describing
the pure skew-morphisms of prime order in an arbitrary cyclic group Zn.
Let n = mp, where p is a prime. Then every element in Zn is written as xm + y
for uniquely defined x ∈ Zp and y ∈ Zm. For b ∈ Z∗p denote by πb the function in
Fun(Zm,Zp) defined as πb(x) = bx.
Proposition 5.2. Let σ be a pure skew-morphism of Zn of order p, p is a prime, and π be
the power function of σ. Then n = pm, (n, p − 1) > 1, and there exist a, b ∈ Z∗p, b 6= 1,
b(n,p−1) = 1 so that
σ(z) = z +ma(∇πb)(y)
π(z) = bz
for all z ∈ Zn and y ∈ Zm, y ≡ z mod (m).
Moreover, every mapping σ : Zpm → Zpm given by the above formula is a pure skew-
morphism of order p and a different choice of the parameters a, b ∈ Z∗p gives different
skew-morphisms.
346 Ars Math. Contemp. 4 (2011) 329–349
Proof. We setG = 〈τ, σ〉. Since σ is pure, by Proposition 3.1 radV (σ) = Cp, Cp ∈ V (σ)
and n = pm for some m. Then σ|m is the identity. The subgroup (Cp)L is centralized by
σ. Therefore σ maps a Zn-element written in the form xm+ y, x ∈ Zp, y ∈ Zm as
σ(xm+ y) = (x+ c)m+ y for some c ∈ Zp.
The value c is the same for x′m + y′ whenever y′ = y, and thus there is a function
ϕ : Zm → Zp which associates c = ϕ(y) with y ∈ Zm. As σ(0) = 0, we have ϕ(0) = 0.
Let σ(1) = am+ 1 ∈ Cp + 1 (see (ii) in Lemma 4.4). Then a 6= 0, and ϕ(1) = a.
By (3.1), στ = τam+1σπ(1). For mx+ y ∈ Zn,
(στ)(xm+ y) =
(
x+ ϕ(y + 1)
)
m+ y + 1,
and also
τam+1σπ(1)(xm+ y) =
(
x+ π(1)ϕ(y) + a
)
m+ y + 1.
Combining these equations we find
ϕ(y + 1) = π(1)ϕ(y) + a for all y ∈ Zm. (5.1)
Applying (5.1) repeatedly leads to
a(1 + π(1) + π(1)2 + · · ·+ π(1)m−1) ≡ 0 (mod p). (5.2)
Since σ /∈ Aut(Zn), π(1) ∈ {2, . . . , p− 1}. Thus (5.2) is equivalent to π(1)m = 1 in
Zp. This means (n, p − 1) > 1, and π(1)(n,p−1) = 1 holds. Inserting b = π(1) into (5.1)
and solving, we get ϕ = a(∇πb). Thus σ has the form as claimed.
It remains to prove that σ is indeed a skew-morphism and to determine its power func-
tion π. It is clear that στm = τσm. We claim that,
στz = τσ(z) σ b
z
for all z ∈ Zn. (5.3)
We prove (5.3) by induction on z. The equality is clear if z = 0 or z = 1. The induction
hypothesis gives
στz+1 = στzτ = τσ(z) σ b
z
τ = τσ(z)σ b
z−1τam+1σb.
Using στam = τamσ and bz = by if z = mx+ y we conclude
στz+1 = τσ(z)+amσ b
z−1τσb
= τσ(z)+2amσ b
z−2τσ2b
· · ·
= τσ(z)+b
zam+1σb
z+1
= τσ(z+1)σb
z+1
.
Comparing (5.3) with (3.1) we see that σ is a skew-morphism of Zn and that the power
function π is given by π(z) = bz , z ∈ Zn.
The formula for |Skew(Zpq)| follows directly from Theorem 5.1 and Proposition 5.2.
Corollary 5.3. If p, q are distinct primes, then
|Skew(Zpq)| =
{
(p− 1)(q − 1) if p 6 | (q − 1) and q 6 | (p− 1)
2(p− 1)(q − 1) if q | (p− 1) or p | (q − 1).
I. Kovács and R. Nedela: Decomposition of skew-morphisms of cyclic groups 347
6 Decomposing skew-morphism of cyclic groups
We are now in a position to prove our decomposition theorem.
Proof of Theorem 1.1. Let σ ∈ Skew(Zn). First, we prove that σ decomposes as σ =
σ1 × σ2, where σi ∈ Skew(Zni) for i = 1, 2. Let A = V (σ). We claim that Cni ∈ A for
i = 1, 2. This we prove by induction on n. If both n1 and n2 are primes, then we are done
by Theorem 5.1.
Let π be the power function of σ. By Corollary 3.2 kerπ 6= 1. Choose Cp ≤ kerπ,
where p is a prime. Thus Cp ∈ A, and Cp is contained either in Cn1 or in Cn2 , say
Cp ≤ Cn1 . Consider the quotient S-ring A/Cp = V (σ|n/p). By the induction hypothesis
we get that Cn1/Cp and Cn2p/Cp are in A/Cp. It follows that Cn1 ∈ A. Next Cn2p/Cp
contains a subgroup of prime order q (p 6= q) which is contained in A/Cp. Consequently,
Cpq ∈ A. By the assumptions p, q are disjoint primes, hence Cq ∈ A, see Theorem 5.1.
Repeating the previous argument with Cq ≤ Cn2 we deduce that also Cn2 ∈ A.
Let us write Zn = Zn1 × Zn2 . Since Cni ∈ A for i = 1, 2, the quotient skew-
morphisms σ|ni , i = 1, 2 are well defined and we have
σ((x1, x2)) =
(
σ|n1(x1), σ|n2(x2)
)
for all (x1, x2) ∈ Zn1 × Zn2 .
Therefore σ = σ|n1 × σ|n2 .
Second, we prove that for all σi ∈ Skew(Zni), i = 1, 2, σ1 × σ2 ∈ Skew(Zn). Let
Gi = 〈τi, σi〉 for i = 1, 2, where τi is the permutation x 7→ x + 1 of Zni . Let Ĝ be
the permutation direct product Ĝ = G1 × G2. Clearly, σ ∈ Ĝ. If both σ1 and σ2 are
automorphisms, then we are done. Assume that σ1 is a pure skew-morphism. Thus there
exists a prime divisor p of n1 such that ord(σ|n1/p) = ord(σ1)/p andCp ∈ V (σ1)⊗V (σ2).
Let B be the block system of Ĝ formed by the Cp-cosets. Then
ĜB = 〈(Cp)L, (σn1/p1 , 1)〉 ∼= Zp × Zp.
Corollary 3.4 gives ord(σ2) | n2φ(n2), and thus p 6 | ord(σ2). We conclude that 〈(σn1/p1 ,
1)〉 is the only subgroup of order p in 〈σ1〉 × 〈σ2〉. We put σ = σ1 × σ2, τ = τ1 × τ2 and
G = 〈τ, σ〉. Then (σn1/p1 , 1) ∈ 〈σ〉 and ĜB ≤ G.
|G|/|ĜB| = |(ĜBG)/ĜB| = |GB| =
∣∣∣〈(τ1 × τ2)B, σB〉∣∣∣.
We have (τ1 × τ2)B = τB1 × τ2 ∼= Zn1/p × Zn2 , and σB = σB1 × σ2 = σ1|n1/p × σ2.
Induction gives σ1|n1/p × σ2 ∈ Skew(Zn/p), hence∣∣∣〈(τ1 × τ2)B, σB〉∣∣∣ = n/p× ord(σ1|n1/p × σ2) = n/p× ord(σ1|n1/p) ord(σ2)(
ord(σ1|n1/p), ord(σ2)
)
As ord(σ1|n1/p) = ord(σ1)/p and p 6 | ord(σ2), we can further write
|G| = |ĜB||GB| = p2 ×
∣∣∣〈(τ1 × τ2)B, σB〉∣∣∣ = n ord(σ1) ord(σ2)(
ord(σ1), ord(σ2)
) .
It follows that G = 〈τ〉〈σ〉 and σ = σ1 × σ2 is a skew-morphism of Zn = 〈τ〉.
348 Ars Math. Contemp. 4 (2011) 329–349
A simple induction leads to the following corollary on the number |Skew(Zn)| of all
skew-morphisms of Zn.
Corollary 6.1. Let n be an odd integer which decomposes into a product of powers of
mutually disjoint primes n =
∏k
i=1 p
ei
i . Then every skew-morphism of Zn is a near auto-
morphism and
|Skew(Zn)| =
k∏
i=1
|Skew(Zni)|.
A number n ≥ 1 is singular if (n, φ(n)) = 1. The following result was proved by
Jones, Nedela and Škoviera in [10].
Theorem 6.2. There is one unique orientably regular embedding of Kn,n if and only if n
is singular.
We conclude the paper with a similar result about arbitrary skew-morphisms of Zn.
Theorem 6.3. All skew-morphisms of Zn are automorphisms of Zn if and only if n = 4 or
n is singular.
Proof. Let n have decomposition n = n1n2 . . . nk such that for all distinct i, j ∈ {1, 2, . . . ,
k}, (ni, nj) = 1, and (ni, ϕ(nj)) = (nj , ϕ(ni)) = 1. Now, all skew-morphisms of Zn are
automorphisms of Zn if and only if |Skew(Zn)| = φ(n). Because of Cororllary 6.1 this
is equivalent to |Skew(Zni)| = φ(ni) for all i ∈ {1, . . . , k}. Proposition 5.2 shows that
|Skew(Zni)| > φ(ni) if ni 6= pei for some prime pi. Thus if n is even, then we must have
n = 2e. It follows readily that |Skew(Z2e)| = φ(2e) if and only if n ≤ 4 (see also the
remark following the proof of Theorem 4.1).
Let n be odd. Then by Proposition 4.9, |Skew(Zni)| = φ(ni) if and only if ni is an
odd prime for all i ∈ {1, . . . , k}, and this means exactly that n is singular.
Theorem 6.2 now becomes an easy corollary of Theorem 6.3.
Acknowledgements
A visit of the first author to the Institute of Mathematics, Slovak Academy of Science in
Banská Bystrica in September 2008 helped us to write the paper. The first author thanks
the Institute of Mathematics for supporting his trip. We would like to thank one of the
anonymous referees for helpful suggestions that have improved both the content and the
presentation of the paper.
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