ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P2.01 / 151–173
https://doi.org/10.26493/1855-3974.2129.ac1
(Also available at http://amc-journal.eu)
Classification of skew morphisms of
cyclic groups which are square roots of
automorphisms*
Kan Hu †
Department of Mathematics, Zhejiang Ocean University,
Zhoushan, Zhejiang 316022, P.R. China, and
Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province,
Zhoushan, Zhejiang 316022, P.R. China
Young Soo Kwon ‡
Department of Mathematics, Yeungnam University,
Gyeongsan, 712-749, Republic of Korea
Jun-Yang Zhang §
School of Mathematical Sciences, Chongqing Normal University,
Chongqing 401331, P.R. China
Received 29 September 2019, accepted 28 April 2021, published online 18 September 2021
Abstract
The auto-index of a skew morphism φ of a finite group A is the smallest positive integer
h such that φh is an automorphism of A. In this paper we develop a theory of auto-index
of skew morphisms, and as an application we present a complete classification of skew
morphisms of finite cyclic groups which are square roots of automorphisms.
Keywords: Skew morphism, auto-index, period, square root.
Math. Subj. Class. (2020): 20B25, 05C10, 14H57
*The authors would like to thank Marston Conder for his suggestion of the concept of ‘auto-index’, and Kai
Yuan for his help in verifying our examples by the Magma program.
†Supported by Natural Science Foundation of Zhejiang Province (LY16A010010, LQ17A010003).
‡Supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education (2018R1D1A1B05048450).
§Corresponding author. Supported by Basic Research and Frontier Exploration Project of Chongqing
(No. cstc2018jcyjAX0010), Science and Technology Research Program of Chongqing Municipal Education Com-
mission (No.KJQN201800512) and National Natural Science Foundation of China (11671276).
E-mail addresses: hukan@zjou.edu.cn (Kan Hu), ysookwon@ynu.ac.kr (Young Soo Kwon),
jyzhang@cqnu.edu.cn (Jun-Yang Zhang)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
152 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
1 Introduction
Throughout the paper, groups considered are all finite. A skew morphism of a group A is
a permutation φ on A fixing the identity element of A and for which there is a function
π : A → Z|φ| on A, called the power function of φ, such that φ(ab) = φ(a)φπ(a)(b) for all
a, b ∈ A. It is apparent the notion of skew morphism is a generalization of that of group
automorphism. A skew morphism of A is called proper if it is not an automorphism. Two
skew morphisms φ and φ′ of A are conjugate if there exists an automorphism θ of A such
that φ′ = θφθ−1.
The concept of skew morphism was first introduced by Jajcay and Širáň in [13] as an
algebraic tool to study regular Cayley maps, which are regular embeddings of graphs on
orientable closed surfaces admitting a regular subgroup of automorphisms on the vertices
of the embedded graph. In this direction, regular Cayley maps of cyclic groups and dihedral
groups have been classified, see [8, 21] and [14, 15, 16, 19, 28, 27]. In contrast, classifi-
cation of regular Cayley maps of non-cyclic abelian groups and other metacyclic groups is
still in progress; see [4, 5, 7, 20, 22, 26] for details.
The connection between skew morphisms and regular Cayley maps reveals a deep re-
lationship between skew morphisms and group factorizations with cyclic complements.
Indeed, if a group G is expressible as a product A⟨y⟩ of a subgroup A and a cyclic sub-
group ⟨y⟩ with A ∩ ⟨y⟩ = 1, then left multiplication of elements of A by y gives rise to a
skew morphism φ of A, determined by ya = φ(a)yπ(a) for all a ∈ A. Conversely, if φ is
a skew morphism of a group A, then for any a, b ∈ A, we have
φLa(b) = φ(ab) = φ(a)φ
π(a)(b) = Lφ(a)φ
π(a)(b),
so ⟨φ⟩LA ⊆ LA⟨φ⟩, where LA = {La | a ∈ A} is the left regular representation of A.
Since ⟨φ⟩∩LA = 1, we have |⟨φ⟩LA| = |LA⟨φ⟩|, and hence ⟨φ⟩LA = LA⟨φ⟩. Therefore,
G = LA⟨φ⟩ is a factorization of a transitive permutation group with a cyclic complement,
which is often referred to as the skew-product group of φ. The interested reader is referred
to [6, 17] for more details.
A prominent problem in this field is the classification of skew morphisms of cyclic
groups, which is closely related to regular Cayley maps [8] as well as edge-transitive
embeddings of complete bipartite graphs [11]. Kovács and Nedela [17] showed that if
n = n1n2 such that gcd(n1, n2) = 1 and gcd(n1, ϕ(n2)) = gcd(ϕ(n1), n2) = 1, then
every skew morphism φ of the cyclic additive group Zn is a direct product φ = φ1×φ2 of
skew morphisms φi of Zni , i = 1, 2. In a subsequent paper [18] the authors classified all
skew morphisms of the cyclic groups Zpe , where p is an odd prime. As for the case p = 2,
the associated skew product groups are classified by Du and Hu in [9].
Recently, Bachratý and Jajcay introduced the notion of period of skew morphisms [1].
More precisely, the period of a skew morphism φ is the smallest positive integer d such
that π
(
φd(a)
)
= π(a) for all a ∈ A. In particular, if d = 1 then the skew morphism
is said to be smooth (or coset-preserving). In [1, 23], it was shown that if φ is a skew
morphism of period d, then φd is a smooth skew morphism. The smooth skew morphisms
of cyclic groups and of dihedral groups were classified in [2] and [23] respectively. Let φ
be a skew morphism of a group A with power function π. If for any a ∈ A either π(a) =
π(φ(a)) = · · · = π(φ|φ|−1(a)) = 1 or π(a) = π(φ(a)) = · · · = π(φ|φ|−1(a)) = t where
|φ| is the order of φ and t is a fixed integer with 1 ≤ t < |φ|, then φ is called t-balanced.
Observe that every t-balanced skew morphism φ of a group A is necessarily smooth, and
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 153
in particular φt+1 is an automorphism of A (see [10] and Remark 3.2 in Section 3). Thus,
any t-balanced skew morphism is a (t+ 1)-th root of a group automorphism.
Inspired by those results above, we propose the following two related problems:
Problem 1.1. Let A be a given group, and d a given positive integer.
(a) Classify all skew morphisms of A which are d-th roots of automorphisms of A.
(b) Classify all skew morphisms of A which have period d.
For A = Zn and d = 2, the following main result of this paper is a solution to the first
problem, and by Theorem 3.8 (a) in Section 4 it is also a partial solution to the second one
(skew morphisms of period 2 of Zn whose square is an automorphism are determined).
Theorem 1.2. Every proper skew morphism of the cyclic additive group Zn which is a
square root of an automorphism is conjugate to a skew morphism of the form
φ(x) ≡ sx− x(x− 1)n
2k
(mod n),
where the pair (k, s) of positive integers satisfy the following conditions:
(a) k2 divides n and s ∈ Z∗n if k is odd, and 2k2 divides n and s ∈ Z∗n/2 if k is even,
(b) s ≡ −1 (mod k), s has multiplicative order 2ℓ in Zn/k and gcd(w, k) = 1 where
w =
k
n
(s2ℓ − 1)− s(s− 1)
2
ℓ.
The power function of φ is given by π(x) ≡ 1+2xw′ℓ (mod m), where w′w = 1 (mod k)
and m = 2kℓ is the order of φ. Moreover, two such skew morphisms corresponding to
distinct integer pairs are not conjugate.
The paper is organized as follows. After a summary of preliminary results in Section 2,
we develop a more comprehensive theory of powers of skew morphisms by defining a new
notion called auto-index in Section 3. In Section 4 we show that if φ is a proper skew
morphism of a group A which is a square root of an automorphism, then its power function
has the property π(xy) ≡ π(x) + π(y) − 1 (mod |φ|) for all x, y ∈ A; in particular, if
A = Zn, then π(x) ≡ (π(1)− 1)x+1 (mod |φ|) for all x ∈ Zn. As an application of the
theory, we present a proof of Theorem 1.2 in Section 5. Finally, for the special case when
n = pe is a prime power, we enumerate proper skew morphisms of Zn which are square
roots of automorphisms in Section 6.
2 Preliminaries
In this section we summarize some preliminary results on skew morphisms for future ref-
erence.
Proposition 2.1 ([1, 13]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. Then for any positive integer k,
φk(ab) = φk(a)φσ(a,k)(b), for all a, b ∈ A,
where σ(a, k) =
k∑
i=1
π(φi−1(a)); moreover, φk is a skew morphism if and only if the
congruence kx ≡ σ(a, k) (mod m) is solvable for every a ∈ A.
154 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
Proposition 2.2 ([13]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. Then for any a, b ∈ A,
π(ab) ≡
π(a)∑
i=1
π(φi−1(b)) (mod m).
Proposition 2.3 ([23]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. Then for any automorphism θ of A,
φ′ = θφθ−1 is a skew morphism of A with power function π′ = πθ−1.
It follows that the automorphism group Aut(A) of A acts by conjugation on the set
Skew(A) of all skew morphisms of A. Two skew morphisms of A are conjugate if they
belong to the same orbit under such action.
An important subgroup related to skew morphisms is the kernel of φ defined by
Kerφ = {a ∈ A | π(a) ≡ 1 (mod m)}.
It is well known that, for any a, b ∈ A, π(a) ≡ π(b) (mod m) if and only if ab−1 ∈ Kerφ,
so π takes exactly |A : Kerφ| distinct values in Zm. The index |A : Kerφ| is called the
skew-type of φ. It is obvious that φ is an automorphism if and only if it has skew-type 1. A
skew morphism which is not an automorphism will be called proper.
The subset
Fixφ = {a ∈ A | φ(a) = a}
of fixed-points of φ forms a subgroup of A. A subgroup N of A is φ-invariant if φ(N) =
N . Clearly, Fixφ is φ-invariant, but Kerφ may not be. However, the subset
Coreφ =
m⋂
i=1
φi(Kerφ)
forms the largest φ-invariant subgroup of A contained in Kerφ, and in particular, it is
normal in A [28]. Thus Kerφ is φ-invariant if and only if Kerφ = Coreφ, in which case
the skew morphism is called kernel-preserving. It is apparent that if φ is kernel-preserving,
then the restriction of φ to Kerφ is an automorphism of Kerφ. The following result is
well known.
Proposition 2.4 ([5]). Every skew morphism of an abelian group is kernel-preserving.
The importance of φ-invariant normal subgroups is reflected by the following result.
Proposition 2.5 ([29]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. If N a φ-invariant normal subgroup of
A, then φ defined by φ(x) = φ(x) is a skew morphism of the quotient group A := A/N . In
particular, the order m1 of φ is a divisor of m, and the power function π of φ is determined
by π(a) ≡ π(a) (mod m1) for all a ∈ A.
Since Coreφ is a normal subgroup of A, φ induces a skew morphism φ of the quotient
group A = A/Coreφ. Define
Smoothφ = {a ∈ A | φ(a) = a},
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 155
which is the preimage of the fixed-point subgroup Fixφ of φ under the natural epimor-
phism of A onto A/Coreφ. Since Fixφ is a φ-invariant subgroup of A, Smoothφ is a
φ-invariant subgroup of A.
In the extremal case that Smoothφ = A, the skew morphism φ is called smooth.
In [23] it is shown that a skew morphism φ of A is smooth if and only if π(a) ≡ π(φ(a))
(mod m) for all a ∈ A. More generally, the period of φ is the smallest positive integer d
such that π(φd(a)) ≡ π(a) (mod m) for all a ∈ A. Thus, φ is smooth if and only if it has
period 1. The following properties on the periodicity of skew morphisms are fundamental,
see [23] for details.
Proposition 2.6 ([23]). Let φ be a skew morphism of a group A, and let π : A → Zm be
the power function of φ, where m is the order of φ. If φ has period d, then the following
hold:
(a) d is equal to the order of the induced skew morphism φ of A = A/Coreφ;
(b) d is the smallest positive integer such that φd is a smooth skew morphism of A;
(c) for any a ∈ A,
d∑
i=1
π(φi−1(a)) ≡ 0 (mod d);
(d) conjugate skew morphisms have identical periods.
Note that for any positive integer k, by Proposition 2.6 (a), if φk is a smooth skew
morphism, then the period d of φ divides k.
3 Skew morphisms and automorphisms
Lemma 3.1. Let φ be a skew morphism of a group A, and let π : A → Zm be the power
function of φ, where m is the order of φ. Then for any positive integer k, φk is a group
automorphism if and only if
k∑
i=1
π
(
φi−1(a)
)
≡ k (mod m)
for all a ∈ A. In particular, if φ is smooth, then φk is an automorphism if and only if
kπ(a) ≡ k (mod m) for all a ∈ A.
Proof. By Proposition 2.1, φk is a skew morphism of A if and only if the congruences
kx ≡ σ(a, k) (mod m) (3.1)
are solvable for all a ∈ A, where
σ(a, k) =
k∑
i=1
π
(
φi−1(a)
)
.
Note that if πµ is the power function of µ := φk, then πµ(a) is the solution of (3.1), and
therefore µ is an automorphism if and only if σ(a, k) ≡ k (mod m) for all a ∈ A. In
addition, if φ is smooth, then σ(a, k) = kπ(a), so µ is an automorphism if and only if
kπ(a) ≡ k (mod m) for all a ∈ A.
156 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
Remark 3.2. If φ is a t-balanced skew morphism of a group A, then φ is smooth and for all
a ∈ A\Kerφ, π(a) ≡ t (mod m) where t2 ≡ 1 (mod m) [5]. Therefore (t+1)t ≡ t+1
(mod m). By Lemma 3.1, φt+1 is a group automorphism. This is a generalization of [10,
Lemma 3.4].
Definition 3.3. For a skew morphism φ of a group A, the auto-index of φ is defined to be
the smallest positive integer h such that φh is a group automorphism of A.
Clearly, φ is an automorphism if and only if it has auto-index 1. Lower and upper
bounds of the auto-index of a skew morphism are given as follows.
Lemma 3.4. Let φ be a skew morphism of a group A. Suppose that φ has order m, period
d and auto-index h, then d divides h and h divides m.
Proof. Note that d is the smallest positive integer such that φd is a smooth skew morphism.
Since φh is an automorphism which is necessarily smooth, the minimality of d implies that
d | h. Since φm = 1 is the identity automorphism, the minimality of h implies that h | m,
as required.
Corollary 3.5. If φ is a proper skew morphism of prime order, then it is smooth with auto-
index equal to its order.
Proof. Let d and h denote the period and auto-index of φ, respectively. As φ is proper,
d ≤ |A : Kerφ| < |φ| and h > 1. By Lemma 3.4, d divides h and h divides |φ|. Since
|φ| = p is prime, we obtain d = 1 and h = p, as required.
As an example of Corollary 3.5, φ = (0)(153)(2)(4) is a proper skew morphism of the
cyclic group Z6. It is smooth, and both its order and auto-index are equal to 3.
Lemma 3.6. Let φ be a skew morphism of the cyclic group Zn and let π : Zn → Zm be
the associated power function, where m is the order of φ. If φ has period 2 and auto-index
h, then h is an even positive divisor of m and there exists some u ∈ Zh such that
π(x) ≡
(
π(1)− 1
) x∑
i=1
(
1 +
um
h
)i−1
+ 1 (mod m), for all x ∈ Zn. (3.2)
Proof. Since φ has period 2, by Proposition 2.6 (c), π(x) + π(φ(x)) ≡ 0 (mod 2) for all
x ∈ Zn. By Lemma 3.4, h is an even positive divisor of m. By Lemma 3.1, we have
h ≡
h∑
i=1
π(φi−1(1)) ≡ 1
2
(
π(1) + π
(
φ(1)
))
h (mod m),
and then
1
2
(
π(1) + π
(
φ(1)
))
= 1 + um/h,
for some u ∈ Zh.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 157
Moreover, since φ has period 2, by Proposition 2.6 (a), φ is an automorphism of order
2. Thus, π(1) ≡ π(1) ≡ 1 (mod 2). Consequently, by Proposition 2.1, we have
π(2) ≡
π(1)∑
i=1
π
(
φi−1(1)
)
≡π(1) + π(1)− 1
2
(
π(1) + π
(
φ(1)
))
≡π(1) +
(
π(1)− 1
)
(1 + um/h)
≡
(
π(1)− 1
)(
1 + (1 + um/h)
)
+ 1 (mod m).
By induction, we obtain (3.2), as required.
In what follows we study skew morphisms of auto-index 2. These skew morphisms are
all square roots of automorphisms. Clearly, every permutation of order 2 on A is a square
root of the identity automorphism of A. Generally, a square root of an automorphism of
A maybe not a skew morphism of A. It seems too difficult to determine all square roots
of automorphisms for a family of groups. In the following example, all square roots of
nonidentity automorphisms of Z8 are determined.
Example 3.7. The cyclic group Z8 has three nonidentity automorphisms as follows:
σ1 = (0)(2)(4)(6)(1, 5)(3, 7), σ2 = (0)(4)(2, 6)(1, 3)(5, 7), σ3 = (0)(4)(2, 6)(1, 7)(5, 3).
Since the square of every permutation of order 4 on Z8 either fixes no element or fixes 4
elements, σ2 and σ3 have no square roots. Set µ = (0)(2)(4)(6)(1, 3, 5, 7) and use Cµ to
denote the set of all square roots of the identity automorphism of Z8 which commute with
µ. Then every square root of σ1 can be represented as a product τµ where τ ∈ Cµ. It is
straightforward to check that µ and µ3 are the only two square roots of σ1 which are skew
morphisms. Since µ3 = σ−13 µσ3, Z8 has a unique conjugate class of skew morphism of
auto-index 2.
We are only concerned with square roots of automorphisms which are also skew mor-
phisms. For convenience, skew morphisms of auto-index 2 are called proper square roots
of automorphisms throughout this paper.
Theorem 3.8. Let φ be a skew morphism of a group A, and let π : A → Zm be the power
function of φ, where m is the order of φ. If φ is a proper square root of an automorphism,
then
(a) φ is kernel-preserving of period at most 2;
(b) π(x) is odd for all x ∈ A;
(c) π(xy) ≡ π(x) + π(y)− 1 (mod m) for all x, y ∈ A;
Proof. Take an arbitrary element x ∈ A. Since φ2 is an automorphism and φ is not an
automorphism, by Lemma 3.1, we have
π(x) + π(φ(x)) ≡ 2 (mod m) and π
(
φ(x)
)
+ π
(
φ2(x)
)
≡ 2 (mod m). (3.3)
158 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
(a) From (3.3) we deduce π(x) ≡ π
(
φ2(x)
)
(mod m), so the period of φ is at most
2. In particular, we see that π
(
φ(x)
)
= 1 whenever π(x) = 1. It follows that φ is kernel-
preserving.
(b) If φ has period 1, then π(x) ≡ π
(
φ(x)
)
(mod m), and hence 2π(x) ≡ π(x) +
π
(
φ(x)
)
≡ 2 (mod m). Since φ is not an automorphism, m must be even. Since π is
a group homomorphism from A to Z∗m [23, Theorem 4.9], π(x) is an odd integer. Now
assume φ has period 2. Since φ is kernel-preserving, Kerφ = Coreφ is normal in A.
By Proposition 2.6 (a), the induced skew morphism φ of A/Kerφ is an automorphism of
order 2. Thus, π(x) ≡ π(x) ≡ 1 (mod 2), and π(x) is also odd.
(c) By Proposition 2.2, we have
π(xy) ≡
π(x)∑
i=1
π(φi−1(y))
≡π(y) + π(x)− 1
2
(
π(y) + π(φ(y))
)
≡π(x) + π(y)− 1 (mod m)
for all x, y ∈ A.
Corollary 3.9. Let φ be a proper square root of an automorphism of a group A, and let
π : A → Zm be the power function of φ, where m is the order of φ. Then
(a) if φ is smooth, then it has skew-type two, 4 divides m, and π(x) = 1 +m/2 for all
x ∈ A \Kerφ;
(b) if φ is not smooth, then it has skew-type at least 3.
Proof. If φ is smooth, then from the proof of Theorem 3.8, we see that m is even and
2π(x) ≡ 2 (mod m) for any x ∈ A. Hence π(x) = 1 or 1 +m/2. Since φ is proper and
π(x) is odd, 4 divides m. If φ is not smooth, then the skew-type of φ is at least 3 since φ
is kernel-preserving of period 2.
Example 3.10 ([25]). The cyclic group Z9 has four skew morphisms of period 2:
φ1 = (0)(1, 2, 7, 5, 4, 8)(3, 6), π1 = [1][3, 5, 3, 5, 3, 5][1, 1];
φ2 = (0)(1, 5, 4, 2, 7, 8)(3, 6), π2 = [1][3, 5, 3, 5, 3, 5][1, 1];
φ3 = (0)(1, 8, 4, 5, 7, 2)(3, 6), π3 = [1][5, 3, 5, 3, 5, 3][1, 1];
φ4 = (0)(1, 8, 7, 2, 4, 5)(3, 6), π4 = [1][5, 3, 5, 3, 5, 3][1, 1].
It can be directly verified that φ2i (i = 1, 2, 3, 4) are automorphisms of Z9, so that all of
these skew morphisms are proper square roots of automorphisms. Note that up to conjuga-
tion by automorphisms they are divided into two classes {φ1, φ4} and {φ2, φ3}.
Example 3.11. Define two functions φ and π on the cyclic group Z8n where n is a positive
integer as follows:
φ(x) ≡
{
2i (mod 8n), if x = 2i;
2(n+ i) + 1 (mod 8n), if x = 2i+ 1
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 159
and
π(x) =
{
1, if x = 2i;
3, if x = 2i+ 1.
It is straightforward to check that φ is a skew morphism of Z8n with power function π
whose square is an involutory automorphism.
4 Technical lemmas
In what follows we restrict our discussion to proper square roots of automorphisms of the
cyclic groups.
Lemma 4.1. Let φ be a skew morphism of the cyclic group Zn, and let π : Zn → Zm be
the power function of φ, where m is the order of φ. If φ is a proper square root of an
automorphism and it has skew-type k, then the following hold:
(a) there is some integer ℓ ≥ 1 such that m = 2kℓ;
(b) there is some integer u ∈ Z∗k such that π(x) ≡ 1 + 2xuℓ (mod m) for all x ∈ Zn;
(c) the number r = φ2(1) is coprime to n and there exists some integer v ∈ Z∗k such
that rℓ ≡ 1 + vn/k (mod n);
(d) k2 is a divisor of n;
(e) the multiplicative order of r in Zn/k is equal to ℓ.
Proof. By Theorem 3.8, φ has period 1 or 2 and
π(x+ y) ≡ π(x) + π(y)− 1 (mod m)
for all x, y ∈ Zn. Thus π(2) ≡ 2π(1)− 1 ≡ 2
(
π(1)− 1
)
+ 1 (mod m) and by induction
π(x) ≡ x
(
π(1)− 1
)
+ 1 (mod m), ∀x ∈ Zn.
In particular, π(m) ≡ m
(
π(1) − 1
)
+ 1 ≡ 1 (mod m), and therefore m ∈ Kerφ. Since
φ is of skew-type k, Kerφ = ⟨k⟩, and hence k | m. Noting that
1 ≡ π(k) ≡ k
(
π(1)− 1
)
+ 1 (mod m),
we get π(1) = 1+um/k for some u ∈ Zk. Consequently, π(x) ≡ 1+xum/k (mod m).
Since π takes k distinct values of the form 1 + im/k (i = 0, 1, . . . , k − 1) in Zm, we
have u ∈ Z∗k. By Theorem 3.8, 1 +m/k is odd, that is, m/k is even. Thus we can write
m = 2kℓ, where ℓ is a positive integer. Then π(x) ≡ 1 + 2xuℓ (mod m).
Set r = φ2(1). Since φ2 ∈ Aut(Zn), r is coprime to n and φ2(x) ≡ rx (mod n) for
all x ∈ Zn. In particular, φ2ℓ(k) ≡ rℓk (mod n). On the other hand, there exists u′ ∈ Zn
such that π(u′) ≡ 1 + 2ℓ (mod m). Therefore
φ(k) + φ(u′) ≡ φ(k + u′) ≡ φ(u′ + k) ≡ φ(u′) + φ1+2ℓ(k) (mod n)
and then φ2ℓ(k) = k. Thus, rℓ ≡ 1 (mod n/k). Write rℓ = 1 + vn/k. Recalling that
φ has period at most 2, we have π
(
φ2ℓ(1)
)
≡ π(1) (mod m) and hence φ2ℓ(1) ≡ 1
160 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
(mod k). It follows that 1 + vn/k ≡ rℓ ≡ φ2ℓ(1) ≡ 1 (mod k), and hence k is a divisor
of vn/k. Note that
φ2ℓj(1) ≡ rℓj ≡
(
1 +
vn
k
)j
≡ 1 + jvn
k
+
j∑
i=2
(
j
i
)(vn
k
)i
≡ 1 + jvn
k
(mod n)
for any positive integer j. By [29, Lemma 3.1], the length of the orbit of 1 under φ is equal
to the order m = 2kℓ of φ. If 0 < j < k, then 1 ̸≡ φ2jℓ(1) ≡ 1 + jvn/k (mod n).
Consequently, v ∈ Z∗k and k2 divides n.
If the multiplicative order of r in Zn/k is i, then ri = 1+tn/k for some positive integer
t. Since rℓ ≡ 1 (mod n/k), we have i | ℓ. On the other hand, since k2 | n for all x ∈ Zn,
we have
φ2ik(x) ≡ rikx ≡ (1 + tn/k)kx ≡ x (mod n).
Since the order of φ is 2kℓ, we get ℓ | i, and therefore ℓ = i.
Corollary 4.2. Let φ be a skew morphism of the cyclic group Zn. If φ is a proper square
root of an automorphism, then the induced skew morphism φ of Zn/Kerφ maps each x to
−x.
Proof. Let m and k be the order and the skew-type of φ, respectively. By Lemma 4.1,
m = 2kℓ for some positive integer ℓ, and
2 ≡ π(x) + π
(
φ(x)
)
≡ 2 + 2
(
x+ φ(x)
)
uℓ (mod 2kℓ)
for all x ∈ Zn, where u ∈ Z∗k. Thus 2
(
x+ φ(x)
)
uℓ ≡ 0 (mod 2kℓ) and then φ(x) ≡ −x
(mod k), as required.
The converse of Corollary 4.2 is generally not true, see [6, Theorem 6.5] for a coun-
terexample. However, we have the following result.
Lemma 4.3. Let φ be a proper skew morphism of the cyclic group Zn. If the induced skew
morphism φ of Zn/Kerφ maps each x to −x, then φ2 is a skew morphism of skew-type at
most 2. In particular, if the skew-type of φ is odd, then φ2 is an automorphism of Zn.
Proof. Throughout the proof, we denote the order and the skew-type of φ by m and k, and
the power functions of φ and φ by π and π, respectively.
If k = 2, then the result is obviously true. In what follows we assume k > 2. Since
φ maps each x to −x, φ is an automorphism of order 2. By Proposition 2.6 (a), φ has
period 2. It follows that m is even, π
(
φ2(x)
)
≡ π(x) (mod m) and π
(
φ(x)
)
≡ π(−x)
(mod m) for all x ∈ Zn. Since π(x) ≡ π(x) ≡ 1 (mod 2), π(x) is odd.
Take two arbitrary elements x, y ∈ Zn. By Proposition 2.2, we have
π(x+ y) ≡
π(x)∑
i=1
π(φi−1(y)) ≡ π(y) + π(x)− 1
2
(
π(y) + π(−y)
)
(mod m).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 161
In particular,
1 = π(x− x) ≡ π(−x) + π(x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.1)
1 = π(−x+ x) ≡ π(x) + π(−x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.2)
π(2x) ≡ π(x) + π(x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.3)
π(−2x) ≡ π(−x) + π(−x)− 1
2
(
π(x) + π(−x)
)
(mod m), (4.4)
π(2x+ 1) ≡ π(2x) + π(1)− 1
2
(
π(2x) + π(−2x)
)
(mod m), (4.5)
π(−2x− 1) ≡ π(−2x) + π(−1)− 1
2
(
π(2x) + π(−2x)
)
(mod m). (4.6)
Adding (4.1) to (4.2) and (4.3) to (4.4), we get
1
2
(
π(x) + π(−x)
)2 ≡ 2 (mod m)
and
1
2
(
π(x) + π(−x)
)2 ≡ π(2x) + π(−2x) (mod m).
Thus,
π(2x) + π(−2x) ≡ 2 (mod m). (4.7)
Substituting 2 for π(2x) + π(−2x) in (4.5) and (4.6) we obtain
π(2x+ 1) ≡ π(2x) + π(1)− 1 (mod m)
and
π(−2x− 1) ≡ π(−2x) + π(−1)− 1 (mod m).
It follows that
π(2x+ 1) + π(−2x− 1) ≡ π(1) + π(−1) (mod m). (4.8)
From (4.7) and (4.8) we deduce that
φ2(x+ y) = φ2(x) + φ2(y)
if x is even, and
φ2(x+ y) = φ2(x) + φπ(1)+π(−1)(y)
if x is odd. Thus, φ2 is a skew morphism of skew-type at most 2. In particular, if the
skew-type k of φ is an odd number, then
π(1) + π(−1) ≡ π(k + 1) + π(k − 1) ≡ 2 (mod m)
and therefore φ2 is an automorphism, as claimed.
162 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
5 Classification
In this section, we classify proper square roots of automorphisms of Zn.
Theorem 5.1. Define a quadratic polynomial over the ring (Zn,+,×) by
φ(x) ≡ sx− x(x− 1)n
2k
(mod n), x ∈ Zn, (5.1)
where k and s are positive integers satisfying the following conditions:
(a) k2 divides n and s ∈ Z∗n if k is odd, and 2k2 divides n and s ∈ Z∗n/2 if k is even,
(b) s ≡ −1 (mod k), s has multiplicative order 2ℓ in Zn/k and gcd(w, k) = 1 where
w =
k
n
(s2ℓ − 1)− s(s− 1)
2
ℓ.
Then φ is a proper square root of an automorphism of the cyclic additive group Zn whose
skew-type is k and power function is given by
π(x) ≡ 1 + 2xw′ℓ (mod m),
where w′w ≡ 1 (mod k) and m = 2kℓ is the order of φ. Moreover, up to conjugation φ
is uniquely determined by the parameters k and s.
Proof. First, we show that φ is a permutation on Zn. Assume φ(x) ≡ φ(y) (mod n)
where x, y ∈ Zn. Then it suffices to prove that x ≡ y (mod n). Since
sx− x(x− 1)n
2k
≡ sy − y(y − 1)n
2k
(mod n),
we get
s(x− y) ≡ (x− y)(x+ y − 1)n
2k
(mod n).
By (a) and (b) we have s ∈ Z∗n. Thus, from the above equation we deduce that x − y ≡ 0
(mod n/k). By (a) again we obtain
(x− y)(x+ y − 1)n
2k
≡ 0 (mod n),
and hence x ≡ y (mod n).
Second, we show that φ2 is an automorphism of Zn. By (a) and (b), we derive from
formula (5.1) that
φ
(jn
k
)
≡ sjn
k
− jn(jn− k)n
2k3
≡ −jn
k
(mod n) (5.2)
for all positive integers j. Now for any x, y ∈ Zn,
φ(x+ y) ≡ s(x+ y)− (x+ y)(x+ y − 1)n
2k
≡ sx− x(x− 1)n
2k
+ sy − y(y − 1)n
2k
− xyn
k
≡ φ(x) + φ(y)− xyn
k
(mod n).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 163
It follows that
φ2(x) ≡ φ
(
sx− x(x− 1)n
2k
)
≡ φ(sx) + φ
(
− x(x− 1)n
2k
)
+
n
k
sx2(x− 1)n
2k
≡ φ(sx) + φ
(
− x(x− 1)n
2k
)
(5.2)
≡ s2x− sx(sx− 1)n
2k
+
x(x− 1)n
2k
≡
(
s2 − s(s− 1)n
2k
)
x− (s
2 − 1)x(x− 1)n
2k
(b)
≡
(
s2 − s(s− 1)n
2k
)
x (mod n).
Since s ∈ Z∗n and k2 | n, we have gcd
(
s2− s(s−1)n2k , n
)
= 1. Thus, φ2 is an automorphism
of Zn.
Next we show that φ is a skew morphism of Zn with associated power function π
defined by π(x) ≡ 1 + 2w′ℓ (mod m) for any x ∈ Zn, where w′w ≡ 1 (mod k). Take
arbitrary x, y ∈ Zn. By the conditions (a) and (b), we have
φ(x) + φπ(x)(y) ≡ φ(x) + φ1+2xw
′ℓ(y) ≡ φ(x) + φ2xw
′ℓ
(
φ(y)
)
≡ φ(x) + φ(y)
(
s2 − s(s− 1)n
2k
)ℓw′x
≡ φ(x) + φ(y)
(
s2ℓ − s(s− 1)ℓn
2k
)w′x
≡ φ(x) + φ(y)
(
1 +
wn
k
)w′x
≡ φ(x) + φ(y)
(
1 +
nx
k
)
(mod n)
and
φ(x+ y) ≡ φ(x) + φ(y)− nxy
k
≡ φ(x) +
(
sy − y(y − 1)n
2k
)
− nxy
k
≡ φ(x) +
(
sy − y(y − 1)n
2k
)
+
snxy
k
≡ φ(x) +
(
sy − y(y − 1)n
2k
)(
1 +
nx
k
)
≡ φ(x) + φ(y)
(
1 +
nx
k
)
(mod n).
Therefore, φ(x+ y) ≡ φ(x) + φπ(x)(y) and thus φ is a skew morphism of Zn.
Finally, we prove that up to conjugation φ is uniquely determined by the parameters k
and s. It is evident that if two such skew morphism are conjugate, then they must have the
same skew-type k. Suppose now that φi (i = 1, 2) are two conjugate skew morphisms of
Zn defined by
φi(x) ≡ six−
x(x− 1)n
2k
(mod n),
164 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
where n, k and si satisfy the stated conditions. Then there exists an automorphism θ of Zn
such that φ1θ = θφ2. Set r = θ(1). Then
s1rx−
rx(rx− 1)n
2k
≡ φ1θ(x) ≡ θφ2(x) ≡ s2rx−
rx(x− 1)n
2k
(mod n).
Since gcd(r, n) = 1, this is reduced to
s1x−
x(rx− 1)n
2k
≡ s2x−
x(x− 1)n
2k
(mod n),
or equivalently,
(s1 − s2)x ≡
x(rx− 1)n
2k
− x(x− 1)n
2k
≡ x
2(r − 1)n
2k
(mod n).
If we choose x = ±1, then ±(s1−s2) ≡ (r−1)n/2k (mod n). Therefore 2(s1−s2) ≡ 0
(mod n) and r ≡ 1 (mod k). If k is even, so is n, and hence s1 ≡ s2 (mod n/2). If both
k and n are odd, then s1 ≡ s2 (mod n). If k is odd but n is even, then r is odd. Since
r ≡ 1 (mod k), we obtain r − 1 ≡ 0 (mod 2k). Thus, we also get s1 ≡ s2 (mod n), as
required.
Now we are ready to prove the main result of the paper.
Proof of Theorem 1.2. By Theorem 5.1, the quadratic polynomial of the stated form is
a proper square root of an automorphism of Zn, and distinct pairs (k, s) correspond to
disconjugate skew morphisms.
Conversely, suppose that φ is a proper square root of an automorphism of Zn of skew-
type k > 1. By Lemma 4.1, k2 | n, |φ| = 2kℓ for some positive integer ℓ, and the power
function of φ is given by π(x) ≡ 1 + 2xuℓ (mod 2kℓ) for some u ∈ Z∗k. Set s = φ(1).
By Lemma 3.1, we have
2 ≡ π(1) + π(φ(1)) ≡ (1 + 2uℓ) + (1 + 2suℓ) ≡ 2 + 2(1 + s)uℓ (mod 2kℓ),
which implies 2(1 + s)uℓ ≡ 0 (mod 2kl). Since u ∈ Z∗k, we obtain s ≡ −1 (mod k).
Since φ2 is an automorphism of Zn, φ2(x) ≡ rx (mod n) for some r coprime to n.
By Lemma 4.1, rℓ ≡ 1 + vn/k (mod n) for some v ∈ Z∗k. Then
φ(x) ≡φ(x− 1) + φπ(x−1)(1) ≡ φ(x− 1) + φ2ℓu(x−1)+1(1)
≡φ(x− 1) + φ2ℓu(x−1)(s) ≡ φ(x− 1) + srℓu(x−1)
≡φ(x− 1) + s
(
1 +
vn
k
)u(x−1)
(mod n).
By induction we obtain
φ(x) ≡ s
x∑
i=1
(
1 +
vn
k
)u(i−1)
(mod n), x ∈ Zn.
Since k2 | n, for any positive integer j, we have
(
1 +
vn
k
)j
≡ 1 + jvn
k
+
j∑
i=2
(
j
i
)(vn
k
)i
≡ 1 + jvn
k
(mod n).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 165
Thus,
φ(x) ≡s
x∑
i=1
(
1 +
vn
k
)u(i−1)
≡ s
x∑
i=1
(
1 +
uvn(i− 1)
k
)
≡s
(
x+
uvnx(x− 1)
2k
)
≡ sx− uvnx(x− 1)
2k
(mod n).
It follows that
r = φ2(1) = φ(s) ≡ s2 − uvns(s− 1)
2k
(mod n). (5.3)
Hence, r ≡ s2 (mod n/k) and by Lemma 4.1 (e), s has multiplicative order 2ℓ in Zn/k.
Since
1 +
vn
k
≡rℓ ≡
(
s2 − s(s− 1)uvn
2k
)ℓ
≡s2ℓ −
(
ℓ
1
)
s2(ℓ−1)
s(s− 1)uvn
2k
+
ℓ∑
i=2
(
ℓ
i
)
s2(ℓ−i)
(
− s(s− 1)uvn
2k
)i
≡s2ℓ − s
2(ℓ−1)s(s− 1)ℓuvn
2k
≡ s2ℓ − s(s− 1)ℓuvn
2k
(mod n),
we have
s2ℓ ≡ 1 +
(
1 +
s(s− 1)ℓu
2
)vn
k
(mod n/k).
By [12, Lemma 1], there exists c ∈ Z∗n such that c ≡ uv (mod k). Define φ′ := θcφθ−1c ,
where θc is the automorphism of Zn taking 1 to c. By Proposition 2.3, φ′ is a skew mor-
phism of Zn. For all x ∈ Zn, we have
φ′(x) =θcφθ
−1
c (x) = θcφ(c
−1x) ≡ c
(
sc−1x− c
−1x(c−1x− 1)cn
2k
)
≡sx− x(x− c)n
2k
≡
(
s+
(c− 1)n
2k
)
x− x(x− 1)n
2k
(mod n).
Let s′ = s + (c−1)n2k , then it is easily seen that s
′ ≡ −1 (mod k), s′ ∈ Z∗n, and s′ has
multiplicative order 2ℓ in Zn/k. Therefore, up to conjugation we can assume
φ(x) ≡ sx− x(x− 1)n
2k
(mod n) and π(x) ≡ 1 + 2w′ℓx (mod 2kℓ),
where s ≡ −1 (mod k), s ∈ Z∗n, w′ ∈ Z∗k, and 2ℓ is the multiplicative order of s in Zn/k.
We show that ww′ ≡ 1 (mod k), that is, w′ is the modular inverse of w in Zk. Noting
that the congruence
w ≡ k
n
(s2ℓ − 1)− s(s− 1)
2
ℓ (mod k)
is equivalent to
s2ℓ − s(s− 1)ℓn
2k
≡ 1 + nw
k
(mod n),
166 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
we have
2s− n
k
≡ φ(2) ≡ φ(1) + φπ(1)(1)
≡ s+ φ2w
′ℓ(s)
≡ s+ s
(
s2 − s(s− 1)n
2k
)ℓw′
≡ s+ s
(
s2ℓ − s(s− 1)ℓn
k
)w′
≡ s+ s
(
1 +
nw
k
)w′
≡ 2s+ sww
′n
k
≡ 2s− nww
′
k
(mod n),
which is reduced to ww′ ≡ 1 (mod k).
In what follows we consider the particular case that k is even. We have
φ2(2) = 2φ2(1) ≡ 2s2 − s(s− 1)n
k
≡ 2s2 − 2n
k
(mod n)
and
φ2(2) ≡ φ
(
2s− n
k
)
≡ s
(
2s− n
k
)
−
(
2s− n
k
)(
2s− n
k
− 1
) n
2k
≡ 2s2 − sn
k
−
(
s− n
2k
)
(2s− 1)n
k
≡ 2s2 − sn
k
−
(
2s2 − s− sn
k
+
n
2k
)n
k
≡ 2s2 − 2s
2n
k
− n
2
2k2
≡ 2s2 − 2n
k
− n
2
2k2
(mod n).
Thus,
2s2 − 2n
k
≡ 2s2 − 2n
k
− n
2
2k2
(mod n),
and therefore 2k2 | n. Moreover, if s > n/2, then we write s′ = s− n/2 and define
φ′(x) ≡ s′x− x(x− 1)n
2k
(mod n), x ∈ Zn.
It is easily seen that φ′ is also a square root of an automorphism of Zn. We show that φ′ is
conjugate to φ. Since 2k2 | n, n = 2ekn1 where e ≥ 1 and 2 ∤ n1. Note that the number
c := kn1 + 1 is coprime to n. Let θc be the automorphism of Zn taking x to cx. Then, for
any x ∈ Zn,
φ′θc(x) ≡ s′cx−
cx(cx− 1)n
2k
≡ (s− n
2
)cx−
(
cx(x− 1) + c(c− 1)x2
)
n
2k
≡ scx− cx(x− 1)n
2k
+
nx
2
− c(c− 1)x
2n
2k
≡ scx− cx(x− 1)n
2k
≡ θcφ(x) (mod n).
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 167
Thus, φ is conjugate to φ′, as required.
Corollary 5.2. Every smooth proper square root of an automorphism of the cyclic group
Zn is conjugate to a skew morphism of the form
φ(x) ≡ sx− x(x− 1)n
4
(mod n), x ∈ Zn,
with the associated power function given by
π(x) ≡ 1 + 2ℓx (mod 4ℓ), x ∈ Zn,
where 8 | n, both s and 2n (s
2ℓ−1)− s(s−1)2 ℓ are odd numbers, and the multiplicative order
of s in Zn/2 is equal to 2ℓ. In particular, φ has order 4ℓ and skew-type 2.
Proof. By Corollary 3.9, every smooth proper square root of an automorphism has skew-
type 2. The result follows immediately from Theorem 1.2.
Remark 5.3. Note that if φ is proper skew morphism of Zn and φ2 is an involutory auto-
morphism, then |φ| = 4, and by Theorem 1.2, k = 2, ℓ = 1 and φ is smooth.
Corollary 5.4. Let φ be a non-smooth skew morphism of the cyclic group Zn. If φ has
skew-type 3, then it is conjugate to a skew morphism of the form
φ(x) ≡ sx− n
6
x(x− 1) (mod n), x ∈ Zn,
where 9 | n, s ∈ Z∗n has multiplicative order 2ℓ in Zn/3, s ≡ −1 (mod 3) and
3
n
(s2ℓ − 1)− ℓ ≡ w′ ̸≡ 0 (mod 3).
Moreover, the order of φ is m = 6ℓ and the power function of φ is given by
π(x) ≡ 1 + m
3
w′x (mod m).
Proof. Since φ is a non-smooth skew morphism of Zn of skew-type 3, the induced skew
morphism φ of Zn/Kerφ is an automorphism of the form φ = (0)(1,−1). By Lemma 4.3,
φ2 is an automorphism. The result then follows from Theorem 1.2.
By Theorem 1.2, we have the following special property of a square root of an auto-
morphism of the cyclic group Zn.
Corollary 5.5. Let φ be a proper square root of an automorphism of the cyclic group Zn.
Then every subgroup of Zn is φ-invariant.
Proof. Let H = ⟨h⟩ be a subgroup of Zn. If φ and φ′ are conjugate by an automorphism
of Zn and H is φ-invariant, then H is also φ′-invariant. So it suffices to consider the skew
morphisms φ given by Theorem 1.2. Let k be the skew-type of φ. For any integer j,
φ(jh) ≡ sjh− jh(jh− 1)n
2k
≡ h
(
sj − j(jh− 1)n
2k
)
(mod n).
If n is even, n2k is a positive integer, and if n is odd, then h is also odd and
j(jh−1)n
2k is a
positive integer. This means that φ(jh) ∈ H , and hence H is φ-invariant.
168 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
6 The prime power case
In this section, for the case where n = pe is a prime power, we enumerate the conjugacy
classes of proper square roots of automorphisms of Zn.
We need a technical result from number theory.
Proposition 6.1 ([3, 24]). Suppose that n = pe, where p is a prime and e ≥ 1. Then
(a) if p > 2, then Z∗pe ∼= Zp−1 × Zpe−1 is cyclic of order pe−1(p− 1). In particular, for
each i, 1 ≤ i ≤ e − 1, an element of the form 1 + upe−i in Z∗pe has order pi if and
only if p ∤ u,
(b) if p = 2, then Z∗2e is trivial if e = 1, Z∗2e ∼= Z2 if e = 2, and Z∗2e ∼= Z2 × Z2e−2 if
e ≥ 3. In particular, in the last case for each i, 2 ≤ i ≤ e − 1, an element of the
form ±1 + u2i in Z∗2e has order 2e−i if and only if 2 ∤ u.
Let N(pe) denote the number of conjugacy classes of proper square roots of automor-
phisms of Zpe . Then N(pe) is determined in the following theorem.
Theorem 6.2. Suppose that p is a prime and e ≥ 1. If p ̸= 2, then
N(pe) =
{
1
p−1 (p
e
2 − 1)2, if e is even
1
p−1 (p
e+1
2 − 1)(p e−12 − 1), if e is odd,
while if p = 2, then
N(2e) =

0, if e < 3
1, if e = 3
2e−1 − 3 · 2 e−22 , if e > 3 is even
2e−1 − 2 e+12 , if e > 3 is odd.
Proof. Denote n = pe and k = pf . Then for fixed prime p and integer e ≥ 1, by Theo-
rem 1.2, N(pe) is equal to the number of pairs (f, s) which satisfy the following conditions:
(a) 2 ≤ 2f ≤ e and s ∈ Z∗pe if p ̸= 2, and 2 ≤ 2f ≤ e− 1 and s ∈ Z∗2e−1 if p = 2,
(b) s ≡ −1 (mod pf ), s has multiplicative order 2ℓ in Zpe−f and p ∤ w, where
w = pf−e(s2ℓ − 1)− 1
2
s(s− 1)ℓ.
For each admissible value of the parameter f , let N(pe, pf ) denote the number of admis-
sible values of the parameter s. In what follows, we first determine N(pe, pf ), and then
determine N(pe). We divide the proof into two cases according to the parity of p.
Case (A). p ̸= 2.
Since s ≡ −1 (mod pf ), we may write s = tph − 1 where 1 ≤ f ≤ h ≤ e and
t ∈ Z∗pe−h . Then s
2 = 1 + tph(tph − 2). According to the multiplicative order 2ℓ of s in
Zpe−f , we distinguish two subcases as follows.
If h < e− f , by Proposition 6.1 we have ℓ = pe−f−h. Since s has multiplicative ordr
2ℓ in Zpe−f , we have pe−f ∥ s2ℓ − 1. Since p | 12s(s− 1)ℓ, we have p ∤ w.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 169
If h ≥ e− f , then ℓ = 1. Recalling that 1 ≤ f ≤ h ≤ e, we have
w ≡ tpf+h−e(tph − 2)− 1
2
(tph − 1)(tph − 2) ≡ −1− 2tpf+h−e (mod p).
Thus, p | w if and only if h = e − f and p | 1 + 2t, where t ∈ Z∗pf , in which case the
number of such t is equal to pf−1.
Consequently,
N(pe, pf ) =
e∑
h=f
ϕ(pe−h)− pf−1 = 1 +
e−1∑
h=f
pe−h−1(p− 1)− pf−1 = pe−f − pf−1,
where ϕ is the Euler’s totient function. Therefore,
N(pe) =
⌊e/2⌋∑
f=1
N(pe, pf ) =
⌊e/2⌋∑
f=1
(pe−f − pf−1) = 1
p− 1
(p⌊e/2⌋ − 1)(pe−⌊e/2⌋ − 1).
Note that ⌊e/2⌋ = e/2 if e is even, and ⌊e/2⌋ = (e− 1)/2 if e is odd. The stated formula
follows from substitution.
Case (B). p = 2.
It is straightforward to check that N(22) = 0, N(23) = N(23, 21) = 1 and N(24) =
N(24, 21) = 2. In what follows, we assume e ≥ 5 and distinguish two subcases.
Subcase (a). s ≡ 1 (mod 4).
Since s ≡ −1 (mod 2f ), we have f = 1. Since s ∈ Z∗2e−1 , we may write s = 1+ 2
ht
where 2 ≤ h ≤ e − 2 and t ∈ Z∗2e−h−1 . By Proposition 6.1 (b), s has multiplicative order
2e−h−1 in Z2e−1 , and so ℓ = 2e−h−2. We have 2 ∤ w since
2e−1 ∥ (s2ℓ − 1) and 2 | 1
2
s(s− 1)ℓ.
Subcase (b). s ≡ −1 (mod 4).
We may write s = −1 + 2ht, where 2 ≤ h ≤ e − 1 and t ∈ Z∗2e−h−1 . Since s ≡ −1
(mod 2f ), we have f ≤ h. Recall that s has multiplicative order 2ℓ in Z2e−f .
If h < e − f − 1, then e > f + h + 1 ≥ 4. By Proposition 6.1, s has multiplicative
order 2e−f−h in Z2e−f , and hence ℓ = 2e−f−h−1. We also have 2 ∤ w since
2e−f ∥ (s2ℓ − 1) and 2 | 1
2
s(s− 1)ℓ.
If h ≥ e− f − 1, then ℓ = 1 and hence
w ≡ 2f−e
(
(−1 + 2ht)2 − 1
)
− (−1 + 2ht)(−1 + 2h−1t)
≡ (−1 + 2h−1t)(2f−e+h+1t− 2ht+ 1)
≡ 2f−e+h+1t+ 1 (mod 2).
It follows that 2 ∤ w if and only if h > e− f − 1. Therefore the case h = e− f − 1 should
be excluded.
170 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
From the above discussion, we obtain
N(2e, 21) =
e−2∑
h=2
ϕ(2e−h−1) +
e−1∑
h=2
ϕ(2e−h−1)− ϕ(2) = 2e−2 − 2,
and for f > 1,
N(2e, 2f ) =
e−f−2∑
h=f
ϕ(2e−h−1) +
e−1∑
h=e−f
ϕ(2e−h−1) = 2e−f−1 − 2f−1.
Consequently, for e ≥ 5, we get
N(2e) =
⌊ e−12 ⌋∑
f=1
N(2e, 2f ) = 2e−2 − 2 +
⌊ e−12 ⌋∑
f=2
(2e−f−1 − 2f−1)
= 2e−2 − 2 + (2⌊
e−1
2 ⌋−1 − 1)(2e−1−⌊
e−1
2 ⌋) − 2).
Note that ⌊ e−12 ⌋ = (e − 2)/2 if e if even, and ⌊
e−1
2 ⌋ = (e − 1)/2 if e is odd. The result
follows from substitution for ⌊ e−12 ⌋ in the above formula, as required.
Remark 6.3. By Theorem 1.2, one can enumerate the conjugacy classes of proper square
roots of automorphisms of Zn for any positive integer n in the following steps:
(a) Find the set of all positive integers k satisfying that k2 divides n if k is odd, and 2k2
divides n if k is even. Denote this set by A(n).
(b) For any k ∈ A(n), find the set of all s satisfying (i) s ≡ −1 (mod k) and (ii) s ∈ Z∗n
if k is odd, and s ∈ Z∗n/2 if k is even. Denote this set by S(n, k).
(c) For any s ∈ S(n, k), calculate the smallest positive integer ℓ such that s2ℓ ≡ 1
(mod n/k) and check whether kn (s
2ℓ − 1)− 12s(s− 1)ℓ is coprime to k or not. Let
A(n, k) be the set of all s ∈ S(n, k) satisfying that kn (s
2ℓ − 1) − 12s(s − 1)ℓ is
coprime to k.
(d) Now (k, s) is admissible for proper square root of automorphism of Zn if and only
if k ∈ A(n) and s ∈ A(n, k). The number N(n) of the conjugacy classes of proper
square roots of automorphisms of Zn is
∑
k∈A(n) |A(n, k)|.
Using the method above, we obtain N(18) = 2, N(24) = 2, N(40) = 2 and N(72) =
16. In each case the parameters (n, k, s) are given below (details are omitted):
(n, k) (18, 3) (24, 2) (40, 2) (72, 2) (72, 3) (72, 6)
s 11, 17 7, 11 11, 19 7, 11, 19, 23, 31, 35 11, 17, 29, 35, 47, 53, 65, 71 23, 35
We close the paper by attaching a full list of conjugacy classes of proper square roots
of automorphisms of Zn for some small values of n.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 171
Table 1: Proper square roots of automorphisms of Zn.
n φ(x) π(x) φ2(x)
8 6x2 + 5x (mod 8) 1 + 2x (mod 4) 5x (mod 8)
9 3x2 + 2x (mod 9) 1 + 2x (mod 6) 4x (mod 9)
9 3x2 + 4x (mod 9) 1 + 2x (mod 6) 4x (mod 9)
16 12x2 + 9x (mod 16) 1 + 2x (mod 4) 9x (mod 16)
16 12x2 + 11x (mod 16) 1 + 2x (mod 4) 9x (mod 16)
18 15x2 + 2x (mod 18) 1 + 2x (mod 6) 13x (mod 18)
18 15x2 + 14x (mod 18) 1 + 2x (mod 6) 7x (mod 18)
24 18x2 + 13x (mod 24) 1 + 2x (mod 4) 23x (mod 24)
24 18x2 + 17x (mod 24) 1 + 2x (mod 4) 13x (mod 24)
27 9x2 + 2x (mod 27) 1 + 6x (mod 18) 4x (mod 27)
27 9x2 + 5x (mod 27) 1 + 6x (mod 18) 25x (mod 27)
27 9x2 + 8x (mod 27) 1 + 2x (mod 6) 10x (mod 27)
27 9x2 + 11x (mod 27) 1 + 6x (mod 18) 13x (mod 27)
27 9x2 + 14x (mod 27) 1 + 12x (mod 18) 7x (mod 27)
27 9x2 + 17x (mod 27) 1 + 4x (mod 6) 19x (mod 27)
27 9x2 + 20x (mod 27) 1 + 6x (mod 18) 22x (mod 27)
27 9x2 + 23x (mod 27) 1 + 12x (mod 18) 16x (mod 27)
32 24x2 + 11x (mod 32) 1 + 4x (mod 8) 25x (mod 32)
32 24x2 + 13x (mod 32) 1 + 4x (mod 8) 25x (mod 32)
32 24x2 + 17x (mod 32) 1 + 2x (mod 4) 17x (mod 32)
32 24x2 + 19x (mod 32) 1 + 4x (mod 8) 9x (mod 32)
32 24x2 + 21x (mod 32) 1 + 4x (mod 8) 9x (mod 32)
32 24x2 + 23x (mod 32) 1 + 2x (mod 4) 17x (mod 32)
32 28x2 + 11x (mod 32) 1 + 2x (mod 8) 9x (mod 32)
32 28x2 + 19x (mod 32) 1 + 6x (mod 8) 25x (mod 32)
40 30x2 + 21x (mod 40) 1 + 2x (mod 4) 31x (mod 40)
40 30x2 + 29x (mod 40) 1 + 2x (mod 4) 21x (mod 40)
64 48x2 + 19x (mod 64) 1 + 8x (mod 16) 41x (mod 64)
64 48x2 + 21x (mod 64) 1 + 8x (mod 16) 25x (mod 64)
64 48x2 + 23x (mod 64) 1 + 4x (mod 8) 17x (mod 64)
64 48x2 + 25x (mod 64) 1 + 4x (mod 8) 17x (mod 64)
64 48x2 + 27x (mod 64) 1 + 8x (mod 16) 25x (mod 64)
64 48x2 + 29x (mod 64) 1 + 8x (mod 16) 41x (mod 64)
64 48x2 + 33x (mod 64) 1 + 2x (mod 4) 33x (mod 64)
64 48x2 + 35x (mod 64) 1 + 8x (mod 16) 9x (mod 64)
64 48x2 + 37x (mod 64) 1 + 4x (mod 16) 57x (mod 64)
64 48x2 + 39x (mod 64) 1 + 4x (mod 8) 49x (mod 64)
64 48x2 + 41x (mod 64) 1 + 4x (mod 8) 49x (mod 64)
64 48x2 + 43x (mod 64) 1 + 8x (mod 16) 57x (mod 64)
64 48x2 + 45x (mod 64) 1 + 8x (mod 16) 9x (mod 64)
64 48x2 + 47x (mod 64) 1 + 2x (mod 4) 33x (mod 64)
64 56x2 + 11x (mod 64) 1 + 12x (mod 16) 25x (mod 64)
64 56x2 + 19x (mod 64) 1 + 4x (mod 16) 9x (mod 64)
64 56x2 + 23x (mod 64) 1 + 2x (mod 8) 17x (mod 64)
64 56x2 + 27x (mod 64) 1 + 12x (mod 16) 57x (mod 64)
64 56x2 + 35x (mod 64) 1 + 4x (mod 16) 41x (mod 64)
64 56x2 + 39x (mod 64) 1 + 6x (mod 8) 49x (mod 64)
172 Ars Math. Contemp. 21 (2021) #P2.01 / 151–173
ORCID iDs
Kan Hu https://orcid.org/0000-0003-4775-7273
Young Soo Kwon https://orcid.org/0000-0002-1765-0806
Jun-Yang Zhang https://orcid.org/0000-0002-0871-2059
References
[1] M. Bachratý and R. Jajcay, Powers of skew-morphisms, in: Symmetries in Graphs, Maps,
and Polytopes, Springer International Publishing, volume 159, pp. 1–25, 2016, doi:10.1007/
978-3-319-30451-9.
[2] M. Bachratý and R. Jajcay, Classification of coset-preserving skew-morphisms of finite cyclic
groups, Australas. J. Comb. 67 (2017), 259–280, https://ajc.maths.uq.edu.au/
?page=get_volumes&volume=67.
[3] B. G. Basmaji, On the ismorphisms of two metacyclic groups, Proc. Amer. Math. Soc. 22
(1969), 175–182, doi:10.2307/2036947.
[4] M. Conder, R. Jajcay and T. Tucker, Regular Cayley maps for finite abelian groups, J. Algebraic
Combin. 25 (2007), 259–283, doi:10.1007/s10801-006-0037-0.
[5] M. Conder, R. Jajcay and T. Tucker, Regular t-balanced Cayley maps, J. Combin. Theory Ser.
B 97 (2007), 453–473, doi:10.1016/j.jctb.2006.07.008.
[6] M. D. E. Conder, R. Jajcay and T. W. Tucker, Cyclic complements and skew morphisms of
groups, J. Algebra 453 (2016), 68–100, doi:10.1016/j.jalgebra.2015.12.024.
[7] M. D. E. Conder, Y. S. Kwon and J. Širáň, Reflexibility of regular Cayley maps for abelian
groups, J. Combin. Theory Ser. B 99 (2009), 254–260, doi:10.1016/j.jctb.2008.07.002.
[8] M. D. E. Conder and T. W. Tucker, Regular Cayley maps for cyclic groups, Trans. Amer. Math.
Soc. 366 (2014), 3585–3609, doi:10.1090/s0002-9947-2014-05933-3.
[9] S. Du and K. Hu, Skew-morphisms of cyclic 2-groups, J. Group Theory 22 (2019), 617–635,
doi:10.1515/jgth-2019-2046.
[10] R. Feng, R. Jajcay and Y. Wang, Regular t-balanced Cayley maps for abelian groups, Discrete
Math. 311 (2011), 2309–2316, doi:10.1016/j.disc.2011.04.012.
[11] Y.-Q. Feng, K. Hu, R. Nedela, M. Škoviera and N.-E. Wang, Complete regular dessins and
skew-morphisms of cyclic groups, Ars Math. Contemp. 18 (2020), 289–307, doi:10.26493/
1855-3974.1748.ebd.
[12] K. Hu, R. Nedela and N.-E. Wang, Nilpotent groups of class two which underly a unique regular
dessin, Geom. Dedicata 179 (2015), 177–186, doi:10.1007/s10711-015-0074-8.
[13] R. Jajcay and J. Širáň, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (2002),
167–179, doi:10.1016/s0012-365x(01)00081-4.
[14] I. Kovács and Y. S. Kwon, Regular Cayley maps on dihedral groups with the smallest kernel,
J. Algebraic Combin. 44 (2016), 831–847, doi:10.1007/s10801-016-0689-3.
[15] I. Kovács and Y. S. Kwon, Classification of reflexible Cayley maps for dihedral groups, J.
Combin. Theory Ser. B 127 (2017), 187–204, doi:10.1016/j.jctb.2017.06.002.
[16] I. Kovács and Y. S. Kwon, Regular Cayley maps for dihedral groups, J. Comb. Theory Ser. B
148 (2021), 84–124, doi:10.1016/j.jctb.2020.12.002.
[17] I. Kovács and R. Nedela, Decomposition of skew-morphisms of cyclic groups, Ars Math. Con-
temp. 4 (2011), 329–349, doi:10.26493/1855-3974.157.fc1.
K. Hu et al.: Classification of skew morphisms of cyclic groups which are square roots . . . 173
[18] I. Kovács and R. Nedela, Skew-morphisms of cyclic p-groups, J. Group Theory 20 (2017),
1135–1154, doi:10.1515/jgth-2017-0015.
[19] J. H. Kwak, Y. S. Kwon and R. Feng, A classification of regular t-balanced Cayley maps on
dihedral groups, European J. Combin. 27 (2006), 382–393, doi:10.1016/j.ejc.2004.12.002.
[20] J. H. Kwak and J.-M. Oh, A classification of regular t-balanced Cayley maps on dicyclic
groups, European J. Combin. 29 (2008), 1151–1159, doi:10.1016/j.ejc.2007.06.023.
[21] Y. S. Kwon, A classification of regular t-balanced Cayley maps for cyclic groups, Discrete
Math. 313 (2013), 656–664, doi:10.1016/j.disc.2012.12.012.
[22] J.-M. Oh, Regular t-balanced Cayley maps on semi-dihedral groups, J. Combin. Theory Ser. B
99 (2009), 480–493, doi:10.1016/j.jctb.2008.09.006.
[23] N.-E. Wang, K. Hu, K. Yuan and J.-Y. Zhang, Smooth skew morphisms of dihedral groups, Ars
Math. Contemp. 16 (2019), 527–547, doi:10.26493/1855-3974.1475.3d3.
[24] M. Xu and Q. Zhang, A classification of metacyclic 2-groups, Algebra Colloq. 13 (2006), 25–
34, doi:10.1142/s1005386706000058.
[25] K. Yuan, Y. Wang and J. H. Kwak, Enumeration of skew-morphisms of groups of small
orders and their corresponding Cayley maps, Adv. Math. (China) 45 (2016), 21–36, doi:
10.1103/physrevd.45.21.
[26] J.-Y. Zhang, Regular Cayley maps of skew-type 3 for abelian groups, European J. Combin. 39
(2014), 198–206, doi:10.1016/j.ejc.2014.01.006.
[27] J.-Y. Zhang, A classification of regular Cayley maps with trivial Cayley-core for dihedral
groups, Discrete Math. 338 (2015), 1216–1225, doi:10.1016/j.disc.2015.01.036.
[28] J.-Y. Zhang, Regular Cayley maps of skew-type 3 for dihedral groups, Discrete Math. 338
(2015), 1163–1172, doi:10.1016/j.disc.2015.01.038.
[29] J.-Y. Zhang and S. Du, On the skew-morphisms of dihedral groups, J. Group Theory 19 (2016),
993–1016, doi:10.1515/jgth-2016-0027.