Skew-morphisms of cyclic <i>p</i>-groups

Kovács, István; Nedela, Roman

doi:10.1515/jgth-2017-0015

Cited by 16 publications

(11 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The coset-preserving skewmorphisms of cyclic groups have been completely determined by Bachratý and Jajcay in [1,2]. In [12], Kovács and Nedela proved that, under certain numerical conditions, a skew-morphism of a cyclic group may be decomposed into a direct product of skew-morphisms of cyclic groups of prime power order; then, in [13], they determined the skew-morphisms of cyclic p-groups where p is an odd prime, leaving the case p D 2 open. In this paper, we shall determine the skew-morphisms of cyclic 2-groups, and thus all skew-morphisms of any cyclic p-group are determined.…”

Section: Motivationsmentioning

confidence: 99%

Skew-morphisms of cyclic 2-groups

2019

Journal of Group Theory

View full text Add to dashboard Cite

A skew-morphism of a finite group A is a permutation φ on A fixing the identity element, and for which there exists an integer function π on A such that, for all {x,y\in A}, {\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)}. In [I. Kovács and R. Nedela, Skew-morphisms of cyclic p-groups, J. Group Theory 20 2017, 6, 1135–1154], Kovács and Nedela determined skew-morphisms of the cyclic p-groups for any odd prime p. In this paper, we shall determine that of cyclic 2-groups.

show abstract

Section: Motivationsmentioning

confidence: 99%

Skew-morphisms of cyclic 2-groups

2019

Journal of Group Theory

View full text Add to dashboard Cite

show abstract

“…For those of Type II, Corollary 6 is helpful. More generally, if both m and n are powers of an odd prime, complete lists of reciprocal pairs of skew-morphisms can be compiled with the helps of the results of Kovács and Nedela [32].…”

Section: Reciprocal Skew-morphismsmentioning

confidence: 99%

Complete regular dessins and skew-morphisms of cyclic groups

Feng¹,

Hu²,

Nedela³

et al. 2018

Preprint

Self Cite

View full text Add to dashboard Cite

A dessin is a 2-cell embedding of a connected 2-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation-and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph K m,n , called (m, n)-complete regular dessins. The purpose is to establish a rather surprising correspondence between (m, n)-complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group A is a bijection ϕ : A → A that satisfies the identity ϕ(xy) = ϕ(x)ϕ π(x) (y) for some function π : A → Z and fixes the neutral element of A. We show that every (m, n)-complete regular dessin D determines a pair of reciprocal skew-morphisms of the cyclic groups Z n and Z m . Conversely, D can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers m and n for which there exists, up to interchange of colours, exactly one (m, n)-complete regular dessin. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order m and n is abelian,

show abstract

“…Subsequently, several articles using skew-morphism as a tool for study of regular Cayley maps were published [4,5,7,11,12,15,16,17,18,20,21,22]. Recently, several papers devoted to the theory of skew-morphisms itself emerged, which focus on cyclic groups [2,8,13,14], dihedral groups [9,19,23] and simple groups [1,3]. In [6], a general theory of skew-morphisms and its connection with complementary products was presented.…”

Section: Introductionmentioning

confidence: 99%

On the direct products of skew-morphisms

Zhang

2022

Art Discrete Appl. Math.

View full text Add to dashboard Cite

A skew-morphism ϕ of a finite group G is a permutation on G fixing the identity element of G and for which there is an integer-valued function π on G such that ϕ(gh) = ϕ(g)ϕ π(g) (h) for all g, h ∈ G. For two permutations α : A → A and β : B → B on the sets A and B, their direct product α × β is the permutation on the Cartesian productIn this paper, necessary and sufficient conditions for a direct product of two skew-morphisms to still be a skew-morphism are given.

show abstract

Skew-morphisms of cyclic p-groups

Cited by 16 publications

References 15 publications

Skew-morphisms of cyclic 2-groups

Skew-morphisms of cyclic 2-groups

Complete regular dessins and skew-morphisms of cyclic groups

On the direct products of skew-morphisms

Contact Info

Product

Resources

About