ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 16 (2019) 527-547 https://doi.org/10.26493/1855-3974.1475.3d3
(Also available at http://amc-journal.eu)
Smooth skew morphisms of dihedral groups*
Na-Er Wang , 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
Kai Yuan
School of Mathematics, Capital Normal University, Beijing 100037, P.R. China
Jun-Yang Zhang
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R. China
Received 4 September 2017, accepted 17 January 2019, published online 28 March 2019
A skew morphism <p of a finite group A is a permutation on A fixing the identity element of A and for which there exists an integer-valued function n on A such that ip(ab) = (fi(a)(f>n(a)(b) for all a, b G A. In the case where n(^(a)) = n(a), for all a G A, the skew morphism is smooth. The concept of smooth skew morphism is a generalization of that of ¿-balanced skew morphism. The aim of this paper is to develop a general theory of smooth skew morphisms. As an application we classify smooth skew morphisms of dihedral groups.
Keywords: Cayley map, skew morphism, smooth subgroup. Math. Subj. Class.: 05E18, 20B25, 05C10
* The authors are grateful to the anonymous referees for their helpful comments and suggestions which have improved the content and presentation of the paper. This research was supported by the following grants: Zhejiang Provincial Natural Science Foundation of China (No. LY16A010010, LQ17A010003); National Natural Science Foundation of China (No. 11801507, 11671276); Teacher Professional Development Program of Zhe-jiang Provincial Education Department (No. FX2017029); Basic Research and Frontier Exploration Project of Chongqing (No. cstc2018jcyjAX0010); Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN201800512); Natural Science Foundation of Fujian (No. 2016J01027).
E-mail addresses: wangnaer@zjou.edu.cn (Na-Er Wang), hukan@zjou.edu.cn (Kan Hu), pktide@163.com (Kai Yuan), jyzhang@cqnu.edu.cn (Jun-Yang Zhang)
Abstract
©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
528
Ars Math. Contemp. 16 (2019) 445-463
1 Introduction
Throughout the paper all groups considered are finite, unless stated otherwise. A skew morphism p of a finite group A is a bijection on the underlying set of A fixing the identity element of A and for which there exists an integer-valued function n: A ^ Z such that p(ab) = p(a)pn(a)(b), for all a, b e A. Note that n is not uniquely determined by p, however, as a permutation if p has order n, then n can be viewed as a function n: A ^ Zn. In this sense the function n is uniquely determined by p, and it will be called the power function of p.
Jajcay and SirM introduced the concept of skew morphism as an algebraic tool to investigate regular Cayley maps [10]. Conder, Jajcay and Tucker have shown in [5] that skew morphisms are also closely related to group factorisations with a cyclic complement. Thus the study of skew morphisms is important for both combinatorics and algebra.
Let X be a generating set of a group A such that 1 e X and X = X -1, let P be a cyclic permutation of X. A Cayley map M = CM(A, X, P) is a 2-cell embedding of the Cayley graph Cay(A, X) into an orientable closed surface such that the local cyclic orientation of the arcs (g, x) emanating from any vertex g induced by the orientation of the supporting surface agrees with the prescribed cyclic permutation P of X .An automorphism of M is an automorphism of the underlying Cayley graph which extends to an orientation-preserving self-homeomorphism of the supporting surface. It is well known that the automorphism group Aut(M) of M acts semi-regularly on the arcs of M .In the case where this action is transitive, and hence regular, the map M is called a regular Cayley map. The left regular representation of A induces a subgroup of map automorphisms which acts transitively on the vertices of M. It follows that M is regular if and only if M admits an automorphism which fixes a vertex, say the identity vertex 1, and maps the arc (1, x) to (1, P(x)). It is a nontrivial result proved by Jajcay and SirM that a Cayley map CM(A, X, P) is regular if and only if there is a skew morphism p of A such that the restriction p \X of p to X is equal to P [10, Theorem 1]. A skew morphism of A will be called a Cayley skew morphism if it has an inverse-closed generating orbit. Thus the study of regular Cayley maps of a group A is equivalent to the study of Cayley skew morphisms of A.
Among the variety of problems considered with regard to skew morphisms the most important seems to be the classification of regular Cayley maps for given families of groups. This problem is completely settled for cyclic groups [6], and only partial results are known for other abelian groups [4, 5, 23]. For dihedral groups Dn of order 2n, if n is odd this problem was solved in [14], whereas if n is even only partial classification is at hand [11,12, 17,21,24,25]. For other non-abelian groups the interested reader is referred to [18,20,21].
Although skew morphisms are usually investigated along with regular Cayley maps, they also deserve to be studied independently in a purely algebraic setting. Let G = AC be a group factorisation, where A and C are subgroups of G with A n C = 1. If C = (c) is cyclic, then the commuting rule ca = p(a)cn(a), for all a e A, determines a skew morphism p of A with the associated power function n. Conversely, each skew morphism p of A determines a group factorisation LA(p) with LA n (p) = 1, where LA denotes the left regular representation of A [5, Proposition 3.1]. Thus, there is a correspondence between skew morphisms and group factorisations with cyclic complements.
Let p be a skew morphism of a group A. A subgroup N of A is p-invariant if p(N) = N. Note that the restriction of p to N is a skew morphism of N, so it is important to study p-invariant subgroups. The first important p-invariant subgroup is Fix p, the subgroup consisting of fixed points of p [10]. Later, Zhang discovered in [25]
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
529
another important (-invariant subgroup, called the core of ( and denoted by Core (. This is a normal subgroup of A, so ( induces a skew morphism ( of the quotient group A := A/ Core ( in a natural way. As a consequence, we obtain a new (-invariant subgroup Smooth ( = {a G A | a G Fix (a} by means of coverings of skew morphisms; see Section 3.
Section 4 is devoted to a study of the extremal case where Smooth ( = A. In this case the skew morphism (is termed smooth. We prove that a skew morphism ( of A is smooth if and only if n(((a)) = n(a) for all a G A. It follows that the power function of a smooth skew morphism takes constant value on orbits of (, so smooth skew morphisms may be viewed as a generalization of ¿-balanced Cayley skew morphisms studied in [4]. Note that for abelian groups smooth skew morphisms are identical with the coset-preserving skew morphisms studied by Bachraty and Jajcay in [1]. We establish in Theorems 4.5 and 4.9 an unexpected relationship between smooth skew morphisms and kernel-preserving skew morphisms. Note that a skew morphism ( of A is kernel-preserving if its kernel Ker (is a (-invariant subgroup of A.
Kovics and Kwon [13] have recently announced a complete classification of regular Cayley maps of dihedral groups. Thus, to complete the classification of skew morphisms of dihedral groups, it remains to determine the non-Cayley skew morphisms. As shown in [8], every non-Cayley skew morphism of dihedral groups is smooth. Our last aim of this paper is to employ the newly-developed theory to give a classification of smooth skew morphisms of the dihedral groups, see Section 5.
2 Preliminaries
In this section we summarize some basic results concerning skew morphisms which will be used throughout the paper.
Let ( be a skew morphism of a group A, let n be the power function of (, and let n be the order of (. As already mentioned above, the sets
Ker ( = {a G A | n(a) = 1}, Fix( = {a G A | ((a) = a}
and
n
Core ( = (®(Ker ()
i=i
form subgroups of A. Note that, for any two elements a, b G A, n(a) = n(b) if and only if ab-1 G Ker (. Thus, the index | A : Ker is equal to the number of distinct values of the power function. This number is called the skew type of (, and it is strictly less than n if ( is not trivial. Clearly, ( is an automorphism of A if and only if it has skew type 1. If ( is not an automorphism, then it will be termed proper. On the other hand, Core ( is the largest (-invariant subgroup contained in Ker (, and in particular, it is normal in A [25].
Lemma 2.1 ([10]). Let ( be a skew morphism of a group A, let n be the power function of and let n be the order of Then, for any a, b G A,
(k(ab) = efc(a)eff(a'fc)(b) and n(ab) = a(b,n(a)) (mod n),
k
where k is an arbitrary positive integer and a(a, k) = n((i-1 (a)).
i= 1
530
Ars Math. Contemp. 16 (2019) 445-463
Lemma 2.2 ([7]). Let f be a skew morphism of a group A, let n be the power function of f. Then for any automorphism y of A, — = y-1(Y is a skew morphism of A with power function = nY. Moreover, Ker — = y-1 (Ker f) and Core — = y-1 (Core f).
Proof. Since y is an automorphism of A, for any a,b e A, we have
—(ab) = y-1(Y (ab) = Y-1f(Y (a)Y (b)) = Y-1 (f (Y(a))f^(a)(Y(b))) = Y-1fY(a)Y-1fn7(a)Y(b) = —(a)—nY(a) (b). Thus, — is a skew morphism of A with power function = nY. Since | —| = |f |, we have
a e Ker — ^^ (a) = 1 (mod |) ^^
nY(a) = 1 (mod |f|) ^^ a e Y-1(Ker f).
Therefore, Ker — = y- 1 (Ker f). Similarly, Core — = y-1 (Core f).	□
Lemma 2.3 ([1, 5]). Let f be a skew morphism of a group A, let n be the power function of f, and let n be the order of f. Then for any positive integer k, ^ = fk is a skew morphism of A if and only if the congruences
kx = a(a, k) (mod n)	(2.1)
are solvable for all a e A. Moreover, if ^ is a skew morphism of A, then it has order m = n/ gcd(n, k) and for each a e A, nM(a) is the solution of the equation (2.1) in Zm.
Lemma 2.4 ([5]). Let f be a skew morphism of a group A. If A is nontrivial, then |f | < |A| and | Ker f | > 1.
Lemma 2.5 ([9]). Let f be a skew morphism of a group A, and let Oa denote the orbit of f containing the element a e A. Then for each a e A, Oa-i = O-1, where O-1 = {g-1 | g e Oa}.
The following result was partially obtained for Cayley skew morphisms in [4].
Lemma 2.6 ([7]). Let f be a skew morphism of a group A, and let n the power function of f, and let n be the order of f. Then for any a e A,
a(a, m) = 0 (mod m),
where m = |Oa| is length of the orbit Oa containing a. Moreover, a(a, n) = 0 (mod n).
Proof. By Lemma 2.1, we have
1 = fm(aa-1) = fm(a)fff(a'm)(a-1) = afff(a'm) (a-1),
so fCT(a'm)(a-1) = a-1. By Lemma 2.5, m = |Oa-i |. Thus, a(a, m) = 0 (mod m). Since m divides n, we obtain
a(a, n) = y^ n(f® 1(a)) = —a(a, m) = 0 (mod n),
m
i=i
as required.	□
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
531
Lemma 2.7 ([7]). Let < be a skew morphism of a group A. Then for any a,b G A, |Oab| divides lcm(|Oa|, |Ob|).
Proof. Denote c = |Oa|, d = |Ob| and i = lcm(|Oa|, |Ob|). Then i = cp = dq for some positive integers p, q. By Lemma 2.1, we have <i(ab) = <i(a)<a(a'i)(b) = a<a(a'e)(b). By Lemma 2.6,
i	c
a(a, i) = ^ n(<i-1(a)) = p ^ n(<i-1(a)) = pa(a,c) = 0 (mod i).
i= 1 i=i
Thus, <i(ab) = ab, and consequently, |Oab| divides i.	□
Lemma 2.8. Let f be a skew morphism of a group A, and let n the power function of f, and let n be the order of f. If A = (a1,..., ar), then n = lcm(|Oai |,..., |Oar |). Moreover, for any g G A, f(g) and n(g) are completely determined by the action of f and the values of n on the generating orbits Oai,..., Oar.
Proof. The first part was first proved in [26, Lemma 3.1]. The reader is invited to give an alternative proof using Lemma 2.7 (and induction on the length of words in the generators).
To prove the second part we use induction on the length k of g in the generators. If k = 1 then g is a generator of A, the assertion is trivially true. Assume that the assertion is true for words of length k. Then, for a word g of length k +1, we have g = ha, where h is a word of length k and a G {ai,..., ar}. By Lemma 2.1, we have
n(h)
f(g) = f(ha) = f(h)fn(h\a) and n(g) = n(ha) = ^ n(fi-1(a)) (mod n).
i= i
Since f(h) and n(h) are completely determined by the action of f and the values of n on the generating orbits, so are f(g) and n(g), as required.	□
Lemma 2.9. Let f be a skew morphism of a group A, let n the power function of f, and let n be the order of f. If N is a f-invariant normal subgroup of A, then
(a)	f induces a skew morphism f of A = A/N by defining f as f(a) = f(a) and the power function n: A ^ Zm associated with f is determined by n(a) = n(a) (mod m) where m = |f |,
(b)	Ker fN/N < Ker f, Core fN/N < Core f and Fix fN/N < Fix f.
Proof. The proof of (a) can be found in [26, Lemma 3.3] while (b) is obvious.	□
3 Invariant subgroups
In this section, we introduce covering techniques to the study of skew morphisms and define several new invariant subgroups.
Proposition 3.1. Let < be a skew morphism of a group A. If M and N are <-invariant subsets of A, so are M n N and MN.
532
Ars Math. Contemp. 16 (2019) 445-463
Proof. For any y G y(M n N), there exists x G M n N such that y = y(x). Since M and N are both y-invariant, y(x) G M and y(x) G N, so y G M n N, whence y(M n N) = M n N. Therefore M n N is also y-invariant. Similarly for any y G ^(MN), there exist u G M and v G N such that y = y(uv). We have y = y(uv) = y(u)yn(u)(v) G y(M)y(N) = MN, so y(MN) = MN, whence MN is also y-invariant.	□
Let n be a finite set of primes, a positive integer k will be called a n-number if all prime factors of k belong to n. For instance, if n = {2,3}, then 2,6,9 are n-numbers, whereas 5,10, 30 are not. We define 1 to be a n-number for any set n of primes.
Now let y be a skew morphism of a group A. An orbit of y will be called a n-orbit if its length is a n-number. Define Orbitn y to be the union of all n-orbits of y, namely,
Orbitn y = {a G A | |Oa| is a n-number}.
Proposition 3.2. Let y be a skew morphism of A, and let n be a finite set of primes, then Orbitn y is a y-invariant subgroup of A containing Fix y.
Proof. By definition, all fixed points of y belong to Orbitn y, so Orbitn y is not empty. Moreover, for any a, b G Orbitn y, |Oa| and |Ob| are n-numbers, so lcm(|Ox|, |Oy |) is also a n-number. By Lemma 2.7, |Oab| divides lcm(|Oa|, |Ob|). It follows that |Oab| is also a n-number. Hence, ab G Orbitn y. Therefore, Orbitn y is a subgroup of A, which is clearly y-invariant.	□
Example 3.3. Consider the skew morphism of the cyclic group Z21 defined by
y = (0) (1, 2,4, 8,16,11) (3, 6,12) (5,10, 20,19,17,13) (7,14) (9,18,15). This is an automorphism of Z21. We have Orbit{2} y = (7), Orbit{3} y = (3), Orbit{5} y = (0), and Orbit{2'3} y = Z21. Now we introduce covering techniques to the study of skew morphisms.
Definition 3.4. Let y» be skew morphisms of finite groups A», i =1, 2. If there is an epimorphism 0: A1 ^ A2 such that the identity
0yi(a) = y20(a)
holds for all a G A1, then y1 will be called a covering (or a lift) of y2, and y2 will be called a projection (or a quotient) of y1. The covering will be denoted by y1 ^ y2, and the epimorphism 0: A1 ^ A2 will be said to be associated with the covering.
Lemma 3.5. Let y1 ^ y2 be a covering between skew morphisms y» of groups A», i = 1,2, and let 0: A1 ^ A2 be the associated epimorphism. Then
(a)	every y1-invariant subgroup M of A1 projects to a y2-invariant subgroup 0(M) of A2,
(b)	every y2-invariant subgroup N of A2 lifts to a y1-invariant subgroup 0-1 (N) of A1.
Proof. (a): For any y G 0(M), y = 0(x) for some x G M. Since M is y1-invariant,
y1(x) G M, so y2(y) = y20(x) = 0y1(x) G 0(M), whence 0(M) is y1-invariant.
(b): For any x G 0-1(N), y = 0(x) G N. Since N is y2-invariant, y2(y) G N, so 0y1(x) = y20(x) = y2(y) G N. Hence y1(x) G 0-1(N).	□
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
533
Since {1}, Fix y2 and Core y2 are all y2-invariant subgroups of A2, by Lemma 3.5, Ker 0 = 0-1(1), 0-1(Fixy2) and 0-1(Core y2) are all y-invariant subgroups of Ai. In particular, both Ker 0 and 0 -1 (Core y2) are normal in A1.
Now we are ready to introduce another new y-invariant subgroup for skew morphisms. Let y be a skew morphism of a group A, and let n be the power function of y. Recall that Core y is a normal y-invariant subgroup of A. Let Smooth y be a subset of A defined by
Smoothy = {a G A | y(a) = a (mod Corey)}.
Proposition 3.6. Let y be a skew morphism of a group A, let n be the power function of y, and let y be the y-induced skew morphism of A = A/ Core y. Then, for any a G A, the following are equivalent:
(a)	a G Smooth y,
(b)	n(y®(a)) = n(a) for all positive integers i,
(c)	a G Fix y.
Proof. (a) (b): Since a G Smooth y, by definition, y(a) = ua for some u G Core y, and so y®(a) = yi-1(u) • • • y(u)ua for all positive integers i. Since Core y is a y-invariant subgroup, we have yi-1(u) • • • y(u)u G Core y. Therefore, n(y®(a)) = n(a).
(b)	(c): Since n(y(a)) = n(a), we have y(a) = ua for some u G Ker y and then y2(a) = y(ua) = y(u)y(a) = y(u)ua. Since n(y2(a)) = n(a), we get y(u)u G Ker y and hence y(u) G Ker y. Repeating the above process, we get y®(u) G Ker y for all positive integers i. Consequently, u G Core y and hence y(l) = a, that is, 4 G Fix y.
(c)	(a): Since 4 G Fix y, we have y(a) = 4 and so y(a) = ua for some u G Core y. Since Core y < A, we obtain a G Smooth y.	□
The following result is a direct corollary of Proposition 3.6.
Corollary 3.7. Suppose that y, A, y and A are defined as Proposition 3.6. Then
Fix y = Smooth y and Smooth y is a y-invariant subgroup of A. In particular,
(a)	Smooth y = Core y ifand only if Fix ya = 1a,
(b)	Smooth y = A if and only if Fix y = A, and
(c)	Smooth y = Fix y if Core y = 1.
Example 3.8 ([22]). Consider a skew morphism of the cyclic group Z18 defined by
y = (0) (1,15,17, 7, 3, 5,13, 9,11) (2,14, 8) (4,10,16) (6) (12), n = [1] [2, 5, 8, 2, 5, 8, 2, 5, 8] [7, 7, 7] [4,4,4] [1] [1].
Then Core y = Ker y = (6), so y = (0) (I, 3, 5) (2) (4) and Smooth y = (2).
The following example is due to Conder, as mentioned in [1],
534
Ars Math. Contemp. 16 (2019) 445-463
Example 3.9. Consider a skew morphism
p = (1) (a, a2) (b, bc, c) (ab, a2bc, ac, a2b, abc, a2c), n = [1] [1,1] [1,4,4] [1,4,4,1, 4, 4]
of the non-abelian group A = D3 x C2, where
D3 = {a,b | a3 = b2 = (ab)2 = 1) and C2 = (c | c2 = 1). We have Ker p = (a, b) and Core p = (a). Thus, p = (I) (b, be, c), and hence
Smooth p = Core p.
4 Smooth skew morphisms
In this section we establish a relationship between kernel-preserving skew morphisms and smooth skew morphisms.
In general, the kernel Ker p of a skew morphism p does not have to be a p-invariant subgroup. However, as we already mentioned above, a skew morphism p will be called kernel-preserving if Ker p is p-invariant. Clearly, p is kernel-preserving if and only if Core p = Ker p. It is well known that every skew morphism p of an abelian group is kernel-preserving [4, Lemma 5.1]. For non-abelian groups, there do exist skew morphisms which are not kernel-preserving, see Example 3.9.
Kernel-preserving skew morphisms have many interesting properties.
Lemma 4.1. Let p be a skew morphism of a group A, n be the power function of p, and let n be the order of p. If p is kernel-preserving, then
(a)	Ker p is a normal subgroup of A, and p restricted to Ker p is an automorphism of Ker p,
(b)	for some positive integer k, if ^ = pk is a skew morphism of A, then Ker p < Ker ^,
(c)	for any automorphism y of A, y-1py is a kernel-preserving skew morphism of A,
(d)	for any pair of elements a e A and u e Ker p there is a unique element v e Ker p such that au = va and p(a)pn(a) (u) = p(v)p(a). In particular, if A is abelian then n(a) = 1 (mod m) where m is the order of the restriction of p to Ker p.
Proof. (a): Since p is kernel-preserving, Ker p = Core p, which is a normal subgroup of A. Moreover, for all a,b e Ker p, we have p(ab) = p(a)p(b), so p restricted to Ker p is an automorphism of Ker p.
(b):	For any a e Kerp = Corep, n(pl-1(a)) = 1, i = 1, 2,... ,n. By Lemma 2.3, the power function of ^ is determined by the the congruence kn^ (a) = a(a, k) = k (mod n), so (a) = 1 (mod n/ gcd(n, k)), which implies that a e Ker ^.
(c):	This is an immediate consequence of Lemma 2.2.
(d):	Since Ker p < A, for any pair (a, u) of elements a e A and u e Ker p, there is a unique element v e Ker p such that au = va. Then p(a)pn(a)(u) = p(au) = p(va) = p(v)p(a). In particular, if A is abelian, then u = v and pn(a) (u) = p(u) for all u e Ker p, so n(a) = 1 (mod m), where m is the order of the restriction of p to Ker p.	□
Proposition 4.2. Every kernel-preserving skew morphism of a non-abelian simple group A is an automorphism of A.
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
535
Proof. If f is not an automorphism of A, then 1 < Ker f < A by Lemma 2.4. Since f is kernel-preserving, by Lemma 4.1(a) Ker f < A, a contradiction.	□
Let f be a skew morphism of a group A. Recall that Smooth f consists of elements a G A such that f(a) = a (mod Core f). If Smoothf = A, then f will be called a smooth skew morphism. The concept of smooth skew morphism was first introduced by Hu in the unpublished manuscript [7]. Bachraty and Jajcay rediscovered it under the name of coset-preserving skew morphisms [1].
Lemma 4.3. Let f be a skew morphism of a group A. If f is smooth, then every subgroup of A containing Core f is f -invariant; in particular, f is kernel-preserving.
Proof. Suppose that f is a smooth skew morphism of A. By Proposition 3.6, the induced skew morphism f of A = A/ Core f is the identity permutation on A, so every subgroup of A is f-invariant. Therefore, by Lemma 3.5, every subgroup of A containing Core f is f-invariant. In particular, since Core f < Ker f, f (Ker f) = Ker f.	□
The following lemma characterizes smooth skew morphisms in terms of their power functions.
Lemma 4.4. Let f be a skew morphism of a group A, and let n be the power function of f. Then f is smooth if and only if n(f (a)) = n(a), for all a G A.
Proof. If f is smooth, then, by Proposition 3.6, n(f (a)) = n(a), for all a G A. Conversely, suppose that, for any a G A, n(f (a)) = n(a). Then f (a) = ua for some u G Ker f. By the assumption, we have n(fn-1(u)) = • • • = n(f (u)) = n(u) = 1, where n = |f |, so u G Core f. Therefore, f (a) = a (mod Core f), that is, f is smooth.	□
The smallest positive integer d such that n(fd(a)) = n(a) (mod |f |), for all a G A, is called the period of f. It is easily seen that d is a divisor of n uniquely determined by f. Bachraty and Jajcay proved that if A is abelian, then p = fd is a smooth skew morphism of A; in particular, if f is nontrivial and contains a generating orbit, then d is a proper divisor of n [1]. In what follows we present a generalization.
Theorem 4.5. Let f be a skew morphism of a group A, let d be the period of f, and let f be the f -induced skew morphism of A = A/ Core f. Then the following hold true:
(a)	d is equal to the order of f,
(b)	a(a, d) = 0 (mod d) for all a G A,
(c)	p = fd is a smooth skew morphism of A,
(d)	p = fd is an automorphism of A if and only if a(a, d) = d (mod n) for all a G A.
Proof. Denote n = |f | and m = |f |.
(a): By the assumption, for any a G A, we have n(fd(a)) = n(a), and so fd(a) = ua for some u G Ker f. Thus,
n(fd+1(a)) = n(f (ua)) = n(f (u)f (a)).
Since n(fd+1(a)) = n(f(a)), we obtain f(u) G Ker f. Repeating this process we get fi-1(u) G Ker f, i = 1,2, ...,n. Thus, u G Core f, and consequently, fd(a) = a.
536
Ars Math. Contemp. 16 (2019) 445-463
Therefore, m < d. On the other hand, since |p| = m, pm(a) = a for any a e A, so pm(a) = ua for some u e Core p. Thus, n(pm(a)) = n(ua) = n(a). The minimality of d then implies that d < m.
(b):	For each a e A, by (a) we have
n	d
a(a,n) = ^n(pi-1(a)) = ^n(p'-1(aj) = ^(a, d) (mod n).
i=1 i=1
By Lemma 2.6, a(a, n) = 0 (mod n) and hence, a(a, d) = 0 (mod d).
(c):	By (b) and Lemma 2.3, ^ = pd is a skew morphism of A with its power function determined by nM(a) = a(a,d)/d (mod n/d). Since n(^(a)) = n(pd(a)) = n(a) (mod n), we obtain a)) = nM(a) (mod n/d). Therefore, ^ is smooth by Proposition 4.4.
(d):	Since nM(a) = a(a,d)/d (mod n/d), ^ is an automorphism if and only if
a(a, d) = d (mod n).	□
Corollary 4.6. Let p be a kernel-preserving skew morphism of a group A, and let n be the order of p. If p is nontrivial, then the period d of p is a proper divisor of n, and so ^ = pd is a nontrivial smooth skew morphism of A.
Proof. If p is nontrivial, then |A : Ker p| < |p| = n. By Lemma 2.4, d = |p| < |A| = | A : Ker p|. Thus, d is a proper divisor of n and therefore, pd is a nontrivial smooth skew morphism by Theorem 4.5.	□
Example 4.7 ([22]). Consider the skew morphism of the cyclic group Z18 given by
p = (0) (1, 5,13,11, 7,17) (2,16, 8,10,14, 4) (3, 5) (6,12) (9), n = [1] [3, 5, 3, 5, 3, 5] [5, 3, 5, 3, 5, 3] [1,1] [1,1] [1].
Then Ker p = Core p = (3) and p = (0) (1,2). Note that p has period 2, which is precisely the order of p. Since a(x, 2) = 0 (mod 2), for all x e Z18, by Theorem 4.5(c), ^ = p2 is an automorphism of A.
Let us revisit the skew morphism p of the non-abelian group D3 x C2 considered in Example 3.9. It has period 3, which is a proper divisor of the order of p. As we already mentioned, the skew morphism is not kernel-preserving. This leads us to pose the following problem.
Problem 4.8. Let d be the period of a nontrivial skew morphism p of a group A. If p is not kernel-preserving, under what condition is ^ = pd nontrivial?
We close this section with some important properties of smooth skew morphisms, see also [1, 7].
Theorem 4.9. Let p be a skew morphism of A, let n be the power function of p, and let n be the order of p. If p is smooth, then
(a)	n: A ^ Z*n is a group homomorphism from A to the multiplicative group Z*n with Ker n = Ker p,
(b)	for any p-invariant normal subgroup N of A, the induced skew morphism p on A/N is also smooth; in particular, if N = Ker p then p is the identity permutation,
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
537
(c)	for any positive integer k, ^ = yk is a smooth skew morphism,
(d)	for any automorphism y of A, — = Y-1yY is a smooth skew morphism of A.
Proof. (a): Since y is smooth, n(a) = n(y(a)) = • • • = n(yn-1(a)). By Lemma 2.1, we have
n(ab) = £ n(yi-1(b)) = n(a)n(b) (mod n).
¿=1
Since 1 = n(ab-1) = n(a)n(a-1) (mod n), n(a) G Therefore, n is a group homo-morphism from A to the multiplicative group
(b):	Since y is smooth, for any a G A, we have n(y(a)) = n(a), and so n(y(a)) = n (a) (mod m), where m = |y|. By Lemma 4.4, y is smooth.
(c):	For any positive integer k, since n(yi-1(a)) = n(a), i = 1, 2,..., k, we have
k
<r(a, k) = ^^n(yi-1(a)) = kn(a) (mod n).
¿=1
It follows that the equations kx = a(a, k) (mod n) are solvable for all a G A. Thus, by Lemma 2.3, ^ = yk is a skew morphism of A and the associated power function
: A ^ Zm is determined by nM(a) = n(a) (mod m), where m = n/ gcd(n, k) is the order of Since nM(^(a)) = n(yk(a)) = n(a) = nM(a) (mod m), by Lemma 4.4, ^ is also smooth.
(d):	By Lemma 2.2, — = Y-1yY is a skew morphism with Core — = Y-1(Core y). For any a G A, since y is smooth, y(Y(a)) = y(a) (mod Core y), or equivalently, Y-1yY(a) = a (mod y-1(Core y)). Thus, -(a) = a (mod Core-0) and hence, — is smooth. □
5 Smooth skew morphisms of dihedral groups
Throughout this section, Dn will denote the dihedral group of order 2n with presentation
Dn = (a, b | an = b2 = 1,b-1ab = a-1), n > 3.	(5.1)
Moreover, for positive integers u and k, t(u, k) and p(u, k) are functions defined by
k k t(u, k) = £ uk-1 and p(u, k) = £(-u)k-1.	(5.2)
¿=1	¿=1
If k is even, we use A(u, k) to denote the function defined by
k/2
A(u,k) = £ u2(i-1).	(5.3)
¿=1
The following result on normal subgroups of Dn is well known.
Lemma 5.1 ([16, Section 1.6, Exercise 8]). Let K be a proper normal subgroup of Dn, n > 3.
(a) if n is odd then K = (a"), where u divides n,
538
Ars Math. Contemp. 16 (2019) 445-463
(b) if n is even, then either K = {a?, b), K = {a?,ab) or K = (au), where u divides n.
Lemma 5.2 ([5]). Let p be a skew morphism of Dn, n > 3, then Ker p = {a).
Lemma 5.3. Let p be a smooth skew morphism of Dn, n > 3. If n is odd, then p is an automorphism of A, whereas if n is even and p is not an automorphism of Dn, then Ker p = {a2), Ker p = {a2,ab) or Ker p = {a2,b). Moreover, the involutory automorphism of Dn taking a ^ a-1,b ^ ab transposes the smooth skew morphisms of Dn with kernels {a?,b) and {a2,ab).
Proof. Assume that p is not an automorphism of Dn, then 1 < Ker p < Dn. Since p is smooth, by Theorem 4.9(a), the power function n: Dn ^ Zf| is a group homomorphism with Ker n = Ker p. It follows that Ker p is a proper normal subgroup of A. Since Z*^ is abelian, D'n < Ker p, where D'n is the derived subgroup of Dn.
If n is odd then D'n = {a), which is a maximal subgroup of Dn. By Lemma 5.2 Ker p = {a), so Ker p = Dn, and hence p is automorphism of Dn, a contradiction.
On the other hand, if n is even, then D'n = {a2), so {a2) < Ker p. By Lemma 5.1, one of the following three cases may happen: Ker p < {a), Ker p = {a2,b), or Ker p = {a2,ab). For the first case, by Lemma 5.2, we have Ker p = {a), so Ker p = {a2).
Finally, by Theorem 4.9(d), the automorphism of Dn taking a ^ a-1,b ^ ab transposes the smooth skew morphisms of Dn with kernels {a2, b) and {a2, ab).	□
The following result classifies smooth skew morphisms of the dihedral groups Dn with
Ker p = {a2) for even integers n > 4.
Theorem 5.4. Let Dn = {a, b) be the dihedral group of order 2n, where n > 4 is an even number. Then every smooth skew morphism p of Dn with Ker p = {a2) is defined by
' p(a2i) = a2iu,	U(a2i) = 1,
p(a2i+!) = a2iu+2r+\	I n(a2i+1) = e,
and	(5.4)
p(a2ib) = a2iu+2sb,	| n(a2ib) = f,	'
Kp(a2i+lb) = a2iu+2r+2sT (u,e)+1b	(n(a2i+1b) = ef,
where r, s, u, e, f are nonnegative integers satisfying the following conditions
(a)	r,s £ Zn/2 and u G Z*n/2,
(b)	the order of p is the smallest positive integer k such that rr (u, k) = 0 (mod n/2) and st(u, k) = 0 (mod n/2),
(c)	e, f £ Z*k generate the Klein four group,
(d)	ue-1 = 1 (mod n/2) and uf= 1 (mod n/2),
(e)	rT(u, e — 1) = u — 2r — 1 (mod n/2) and st(u, f — 1) = 0 (mod n/2),
(f)	rT(u, f — 1) + st(u, e — 1) = u — 2r — 1 (mod n/2).
Proof. First suppose that p is a smooth skew morphism of Dn with Ker p = {a2). Then by Theorem 4.9(b), the induced skew morphism p on Dn/ Ker p is the identity permutation, so there exist integers r,s G Zn/2 such that
2s i
p(a) = a1+2r and p(b) = a2s b.
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
539
Since ^ is kernel-preserving, the restriction of ^ to Ker ^ = (a2} is an automorphism, so
^>(a2) = a2" where u G Z^/2. Assume that n(a) = e (mod k) and n(b) = f (mod k), where k = |<^|.
From the above identities we derive the following formulae by induction:
(a) = a1+2rT ("j) and	(b) = a2sT ("j)b,
where j is a positive integer and t(u, j) = Xj=1 ui-1. Since Dn = (a, b}, by Lemma 2.8, the order k = |y>| is equal to lcm(|Oa|, |Ob|), the least common multiple of the lengths of the orbits containing a and b. That is, k is the smallest positive integer such that (a) = a and (b) = b. Using the above formulae we then deduce that k is the smallest positive integer such that tt(u, k) = 0 (mod n/2) and st(u, k) = 0 (mod n/2).
Now we determine the skew morphism and the associated power function. By the assumption we have
^(a2i) = (a2")4 = a2i", ^(a2ib) = ^(a2i)^(b) = a2i"+2sb.
Similarly, we have
^(a2i+1) = ^(a2ia) = ^(a2i)^(a) = a1+2r+2i", ^(a2i+1 b) = ^(a2i)^(a)^e(b) = a2i"+1+2r+2sT ("'e).
Since n: Dn ^ Zk is a group homomorphism, we have e2 = n(a)2 = n(a2) = 1 (mod k) and f2 = n(b)2 = n(b2) = 1 (mod k), so e2 = 1 (mod k) and f2 = 1 (mod k). Hence, n(a2i) = 1, n(a2i+1) = e, n(a2ib) = f, n(a2i+1b) = ef. In particular, since |Dn : Ker =4, (e, f} < Zk is the Klein four group. Therefore ^ and n have the claimed form (5.4). Moreover, we have
a1+2r+2"e = ^(a)^e(a2) = ^(a)^n(a)(a2) = ^(aa2) = ^(a2a) = ^(a2)^(a) = a1+2r+2",
and so ue-1 = 1 (mod n/2). Similarly, since
(a2) = ^(b)^n(6)(a2) = ^(ba2) = ^(a-2b) = ^(a-2)^(b),
we have
a2s-2"f b = a2sba2"f =	(a2) = ^(a-2)^(b) = a2s-2"b.
Thus, uf-1 = 1 (mod n/2). Furthermore, since
a2" = ^(a2) = ^(a)^n(a)(a) = ^(a)^e(a) = a2+2r+2rT ("'e),
we get
t(1+ t(u, e)) = u - 1 (mod n/2).	(5.5)
540
Ars Math. Contemp. 16 (2019) 445-463
Similarly,
1 = y(b2) = y(b)yn(b)(b) = y(b)yf (b) = a2sba2sr ^ = a2s-2sr we obtain
st(u, f ) = s (mod n/2).	(5.6)
Employing induction it is easy to deduce that yj(a-1) = a1-2"3+2rr("j), where j is an arbitrary positive integer. Then
y(a)ye(b) = y(ab) = y(ba-1) = y(b)yf (a-1).
Upon substitution we get
a1+2r+2sr ("'e)b = y(a)ye (b) = y(b)yf (a-1) = a2sba1-2"f+2rr )
= a2s-1+2uf -2rr(«,/)b
Hence,
rT(u, f) + st(u, e) = s + uf — r — 1 (mod n/2). Since uf = u (mod n/2), the congruence is reduced to
rT(u, f ) + st(u, e) = s + u — r — 1 (mod n/2).	(5.7)
Recall that ue-1 = 1 (mod n/2) and uf-1 = 1 (mod n/2), so
t(u, e) = t(u, e — 1) + 1 (mod n/2), t(u,f) = t(u,f — 1) + 1 (mod n/2).
Upon substitution the congruences (5.5), (5.6) and (5.7) are reduced to the numerical conditions in (e) and (f).
Conversely, for a quintuple (r, s, u, e, f ) of nonnegative integers satisfying the stated numerical conditions, we verify that y given by (5.4) is a smooth skew morphism of Dn with Ker y = (a2) and the function n is the associated power function. It is evident that y is a bijection on Dn and y(1) = 1.
It remains to verify the identity y(xy) = y(x)yn(x)(y) for all x, y G Dn. By Lemma 2.8, it suffices to verify this for x, y G Oa U Ob, where Oa and Ob are the generating orbits of y of the form
Oa = (a, a1+2rr("'1), a1+2rr("'2),. .., a1+2rr("'i),. ..), Ob = (b, a2sr("'1)b, a2sr("'2)b,. .., a2sr("j)b,. ..).
It follows that one of the following four cases may happen:
(i)	x, y G Oa;
(ii)	x, y G O6;
(iii)	x G Oa, y G Ob or
(iv)	x G O6, y G Oa.
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
541
We shall demonstrate the verification for the first case, and leave other cases to the reader.
If x, y € Oa, then x = a1+2rr(u,i) and y = a1+2rr("'j) for some i, j. We have
y(xV(y) = ^(a2r(r ("'i)+T ("j))+2) = a2ru(r ("'i)+T (uj))+2"
and
^(x)^n(x)(y) = ^(a1+2rr ("'i) Ve(a1+2rr ("j)) = a2r(r ("■i+1)+T (uj+e))+2. By the numerical conditions (d) and (e), we have
r(r(u, i + 1) + t(u, j + e)) + 1 - (ru(r(u, i) + t(u, j)) + u)
= r ^(t(u, i +1) — ut(u, i)) + (t(u, j + e) — ut(u, j))^ +1 — u
= 1 + (t (u, j + e) — ueT (u, j ))j +1 — u
= r(2 + t(u, e — 1)) + 1 — u
(e)
= 0 (mod n/2).
Therefore, y(xy) = ^(x)^n(x)(y).
Finally, from the choices of the parameters it is easily seen that distinct quintuples (r, s, u, e, f) give rise to different skew morphisms of Dn, as required.	□
Remark 5.5. In Theorem 5.4, consider the particular case where u = 1. By Condition (b) we have
(gcd(r, n/2), gcd(s, n/2)) The numerical conditions are reduced to
V(e + 1) = 0 (mod n/2), s(f — 1) = 0 (mod n/2), ,r(f + 1) + s(e — 1) = 0 (mod n/2),
where r, s € Zn/2 and (e, f} < Z^ is the Klein four group. If n = 8m, where m > 3 is an odd number, then it can be easily verified that the quintuple (r, s,u, e, f) = (m + 4, m, 1,4m — 1, 2m — 1) fulfills the numerical conditions. Therefore, we obtain an infinite family of skew morphisms of D8m of order 4m with Ker ^ = (a2}. This example was first discovered by Zhang and Du in [26, Example 1.4].
Example 5.6. By computations using the Magma system we found that the smallest n for which there is a smooth skew morphism ^ of Dn with Ker ^ = (a2} is the number 24. In this case, all such skew morphisms have order 12, and the corresponding quintuples
(r, s, u, e, f) are listed below:
(r, s, u, e, f) = (1, 3,1,11, 5), (1,4,1,11, 7), (1, 9,1,11, 5), (1,10,1,11, 7), (5, 2,1,11, 7), (5, 3,1,11, 5), (5, 8,1,11, 7), (5, 9,1,11, 5), (7, 3,1,11, 5), (7,4,1,11, 7), (7, 9,1,11, 5), (7,10,1,11, 7), (11,, 2,1,11, 7), (11, 3,1,11, 5), (11, 8,1,11, 7), (11, 9,1,11, 5).
542
Ars Math. Contemp. 16 (2019) 445-463
Note that in each case we have u = 1, so the restriction of p to Ker p is the identity automorphism of Ker p. However, further computations show that, for other n, there do exist examples with u =1.
For even numbers n, by Lemma 5.3, the involutory automorphism 7 of Dn taking a ^ a-1, b ^ ab transposes the smooth skew morphisms of Dn with kernels (a2, b) or (a2 ,ab). Thus, to complete the classification of smooth skew morphisms of Dn, it suffices to determine the smooth skew morphisms of Dn with kernel Ker p = (a2,b).
Theorem 5.7. Let Dn be the dihedral group of order 2n, where n > 8 is an even number. If p is a smooth skew morphism of Dn with Ker p = (a2 ,b), then p belongs to one of the following two families of skew morphisms:
(I) skew morphisms of order k defined by
-,2 iu
2iu+2r + l
p(a) = p(a2i+1)= a p(ba2i) = ba2iu+2s, Kp(ba2i+1) = ba2r+2s+2iu+1
and
' n(a2i) = 1, n(a2i+1) = e, n(ba2i) = 1, kn(ba2i+1) = e,
(5.8)
where r, s, u, k, e are nonnegative integers satisfying the following conditions
(a)	r,s G Zn/2 and u G Z*n/2,
(b)	k is the smallest positive integer such that rr (u, k) = 0 (mod n/2) and st(u, k) = 0 (mod n/2),
(c)	e G Z*k such that e ^ 1 (mod k) and e2 = 1 (mod k),
(d)	ue-1 = 1 (mod n/2),
(e)	rT(u, e — 1) = u — 2r — 1 (mod n/2) and st (u, e — 1) = — u + 2r + 1 (mod n/2).
(II) skew morphisms of order 2(e — 1) defined by
p(a2i) = a2
p(a2i+1) = ba2r-2iu+1, p(ba2i) = ba2s+2iu, p(ba2i+1) = a2r-2s-2iu+1
and
n(a2i) = 1, n(a2i+1) = e, n(ba2i ) = 1, ,n(ba2i+1 ) = e,
(5.9)
where r, s, u, e are nonnegative integers satisfying the following conditions
(a)	r,s G Zn/2, u G Z*n/2 and e > 1 is an odd number,
(b)	ue-1 = -1 (mod n/2),
(c)	st(u, e — 1) = u + 2r +1 (mod n/2),
(d)	rp(u, e — 1) = sA(u, e — 1) — 1 (mod n/2).
Proof. First suppose that p is a smooth skew morphism of Dn with Ker p = (a2, b). By Theorem 4.9, the induced skew morphism p of Dn / Ker p is the identity permutation and
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
543
the restriction of y to Ker y = (a2, b) is an automorphism of Ker y. It follows that there exist integers r, s, u G Zn/2 and I G Z2 such that
y(a) = bV+2r, y(b) = ba2s and y(a2) = a2u.
Assume that n(a) = e (mod k), where k = |y| denotes the order of y. Since b G Ker y, n(b) = 1 (mod k). By Theorem 4.9, the power function n: Dn ^ Zk is a group homomorphism from Dn to the multiplicative group Z£, so
e-1 = n(a-1) = n(b-1ab) = n(a) = e (mod k),
and hence e2 = 1 (mod k). It follows that n(a2i) = n(a2ib) = 1 and n(a2i+1) = n(a2i+1b) = e. Since y has skew type 2, e = 1 (mod k). To proceed we distinguish two cases:
Case (I): I = 0.
In this case, we have
Then
Similarly,
y(a) = a1+2r, y(b) = ba2s and y(a2) = a2u.
y(a2i) = y(a2)4 = a2iu, y(ba2i) = y(b)y(a2)® = ba2iu+2s.
y(a2i+1) = y(a2ia) = y(a2)V(a) = a2iu+2r+1, y(ba2i+1) = y(ba2ia) = y(b)y(a2i)y(a) = ba2r+2s+2i"+1.
Hence, the skew morphism has the form given by (5.8). Using induction it is easy to prove that
yj (a) = a1+2rT	and yj (b) = ba2sT
where j is a positive integer and t(u, j) = ^j=1 ui-1. Since Dn = (a, b), k = |y| is the smallest positive integer such that yk (a) = a and yk (b) = b, which implies that
rT(u, k) = 0 (mod n/2) and st(u, k) = 0 (mod n/2).
Moreover, we have
a1+2r+2"e = y(a)ye(a2) = y(aa2) = y(a2a) = y(a2)y(a) = a1+2r+2",
so ue 1 = 1 (mod n/2). Furthermore, since
a2u = y(a2) = y(a)ye(a) = a1+2r a1+2rT = a2+2r+2rT , we obtain
r(t(u,e) + 1) = u - 1 (mod n/2).	(5.10)
544
Ars Math. Contemp. 16 (2019) 445-463
Similarly,
p(a)pe(b) = p(ab) = p(ba-1) = p(b)p(a-1) = p(b)p(a-2a) = p(b)p(a-2)p(a). By the above formula we have
p(a)pe(b) = a1+2r ba2sT ("'e) = ba-1-2r+2sT ("'e)
and
p(b)p(a-2)p(a) = ba1+2r+2s-2".
Consequently, upon substitution we obtain
s(t(u, e) — 1) = — u + 2r +1 (mod n/2).
Recall that ue-1 = 1 (mod n/2), so
t(u, e) = t(u, e — 1) + ue-1 = t(u, e — 1) + 1 (mod n/2).
Upon substitution the equations (5.10) and (5.11) are reduced to
rT(u, e — 1) = u — 2r — 1 (mod n/2), st(u, e — 1) = —u + 2r +1 (mod n/2).
Case (II): I = 1. In this case we have
Then
p(a) = ba1+2r, p(b) = ba2s and p(a2) = a2u.
p(a2i) = a2iu, p(ba2i) = p(b)p(a2i) = ba2s+2iu.
Similarly,
p(a2i+1) = p(a2ia) = a2i"ba1+2r = ba2r-2iu+1 p(ba2i+1) = p(b)p(a2i)p(a) = a2r-2s-2i"+1.
Hence p has the form (5.9).
Using induction it is easy to derive the following formula
(5.11)
p
j (b)= ba2sT ("'j) and pj (a) =
2rp(«,j)-2sA(«,j) + 1
if j is even,
ba2rp("'j)+2s"A("'j-1)+1, if j is odd,
where t, p and A are the functions defined by (5.2) and (5.3). Since p(a) = ba1+2r and
Dn = (a, ba1+2r), k = |p| = |Oa |. Thus, k is the smallest positive integer such that
rp(u, k) = sA(u, k) (mod n/2).
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
545
In particular, since elements from the cosets (a) and b(a) alternate in the orbit Oa, k is even, and hence e is odd. Thus,
a2u = ((a2) = ((a)(e (a) = ((a)(e (a) = a2rP(u'e)-2r+2suX(u'e-1).
Consequently, we obtain
rp(u, e) + suA(u, e — 1) = r + u (mod n/2).	(5.12)
Furthermore, we have
ba1+2r+2"e = ((a)(e(a2) = ((aa2) = ((a2a)
= ((a2)((a) = a2"ba1+2r = ba2r-2u+1,
so ue-1 = —1 (mod n/2). Similarly
a-1-2r+2sT ("'e) = ((a)(e(b) = ((ab) = ((ba-2a) = ((b)((a-2)((a) = a1+2r-2s+2u.
Hence
st(u, e) = 1 + 2r + u — s (mod n/2).	(5.13)
Recall that ue-1 = —1 (mod n/2), so
t(u, e) = t(u, e — 1) — 1 (mod n/2), p(u, e) = p(u, e — 1) — 1 (mod n/2).
Upon substitution the equations (5.12) and (5.13) are reduced to
rp(u, e — 1) + suA(u, e — 1) = 2r + u (mod n/2),	(5.14)
st(u, e — 1) = 2r + u +1 (mod n/2).	(5.15)
Subtracting we then get
rp(u, e — 1) = sA(u, e — 1) — 1 (mod n/2).
Finally, note that
2(e-1)
p(u, 2(e — 1))= £ (—u)2(e-1) i=1
e- 1	e- 1
=	+ ^E(-u)i-1 = 0 (mod n/2),
i=i	i=i
and
e-1
u2i
A(u, 2(e - 1)) = £>
i=i
(e-1)/2	(e-1)/2
= ^ u2(i-1) + ue-1 ^ u2(i-1) = 0 (mod n/2),
i=1 i=1
546
Ars Math. Contemp. 16 (2019) 445-463
Hence,
rp(u, 2(e - 1)) = sA(u, 2(e - 1)) (mod n/2). The minimality of k yields k | 2(e — 1). But e — 1 < k, which forces k = 2(e — 1).
Conversely, in each case for any quadruple (r, s, u, e) satisfying the numerical conditions, it is straightforward to verify that f of the given form is a smooth skew morphism of Dn with Ker f = (a2, b) and n is the associated power function. In particular, from the choices of the parameters it is easily seen that distinct quadruples (r, s, u, e) give rise to different skew morphisms of Dn, as required.	□
Remark 5.8. Let f be any skew morphism from (II) of Theorem 5.7. Note that the orbit of f containing a also contains ba2r+1, so the orbit Oa generates Dn. Clearly, it is closed under taking inverses. Therefore, such skew morphisms give rise to the e-balanced regular Cayley maps of Dn, which were first classified by Kwak, Kwon and Feng [17].
References
[1]	M. Bachraty and R. Jajcay, Powers of skew-morphisms, in: J. Sirdn and R. Jajcay (eds.), Symmetries in Graphs, Maps, and Polytopes, Springer, Cham, volume 159 of Springer Proceedings in Mathematics & Statistics, 2016 pp. 1-25, doi:10.1007/978-3-319-30451-9_1, papers from the 5th SIGMAP Workshop held in West Malvern, July 7 - 11, 2014.
[2]	M. Bachraty and R. Jajcay, Classification of coset-preserving skew-morphisms of finite cyclic groups, Australas. J. Combin. 67 (2017), 259-280, https://ajc.maths.uq.edu.au/ pdf/67/ajc_v67_p25 9.pdf.
[3]	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.
[4]	M. Conder, R. Jajcay and T. Tucker, Regular ¿-balanced Cayley maps, J. Comb. Theory Ser. B 97 (2007), 453-473, doi:10.1016/j.jctb.2006.07.008.
[5]	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.
[6]	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.
[7]	K. Hu, Theory of skew morphisms, 2012, preprint.
[8]	K. Hu and Y. S. Kwon, Regular Cayley maps and skew morphisms of dihedral groups: a survey, in preparation.
[9]	R. Jajcay and R. Nedela, Half-regular Cayley maps, Graphs Combin. 31 (2015), 1003-1018, doi:10.1007/s00373-014- 1428-y.
[10]	R. Jajcay and J. Sirffi, Skew-morphisms of regular Cayley maps, Discrete Math. 244 (2002), 167-179, doi:10.1016/s0012-365x(01)00081-4.
[11]	I. Kovdcs 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.
[12]	I. Kovdcs and Y. S. Kwon, Classification of reflexible Cayley maps for dihedral groups, J. Comb. Theory Ser. B 127 (2017), 187-204, doi:10.1016/j.jctb.2017.06.002.
[13]	I. Kovdcs and Y. S. Kwon, private communication, 2018.
[14]	I. Kovdcs, D. Marusic and M. Muzychuk, On G-arc-regular dihedrants and regular dihedral maps, J. Algebraic Combin. 38 (2013), 437-455, doi:10.1007/s10801-012-0410-0.
N.-E. Wang et al.: Smooth skew morphisms of dihedral groups
547
[15]	I. Kovdcs 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.
[16]	H. Kurzweil and B. Stellmacher, The Theory of Finite Groups: An Introduction, Universitext, Springer-Verlag, New York, 2004, doi:10.1007/b97433.
[17]	J. H. Kwak, Y. S. Kwon and R. Feng, A classification of regular ¿-balanced Cayley maps on dihedral groups, European J. Combin. 27 (2006), 382-393, doi:10.1016/j.ejc.2004.12.002.
[18]	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.
[19]	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.
[20]	J.-M. Oh, Regular t-balanced Cayley maps on semi-dihedral groups, J. Comb. Theory Ser. B 99 (2009), 480-493, doi:10.1016/j.jctb.2008.09.006.
[21]	Y. Wang and R. Q. Feng, Regular balanced Cayley maps for cyclic, dihedral and generalized quaternion groups, Acta Math. Sin. 21 (2005), 773-778, doi:10.1007/s10114-004-0455-7.
[22]	K. Yuan, Y. Wang and J. H. Kwak, Enumeration of skew-morphisms of cyclic groups of small orders and their corresponding Cayley maps, Adv. Math. (China) 45 (2016), 21-36.
[23]	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.
[24]	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.
[25]	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.
[26]	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.