Half-Regular Cayley Maps

Jajcay, Robert; Nedela, Roman

doi:10.1007/s00373-014-1428-y

Cited by 8 publications

(4 citation statements)

References 10 publications

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“…The next three properties of skew-morphisms are well known and were proved in [18,19,42]. For convenience, we include the proof of the third of them.…”

Section: Reciprocal Skew-morphismsmentioning

confidence: 89%

“…Thus the classification of regular Cayley maps of a finite group A is essentially a problem of determining certain skew-morphisms of A. Since then, the theory of skew-morphism has become a dispensable and powerful tool for the study of regular Cayley maps; the interested reader is referred to [5,4,18,28,29,33,30,34,35,43,44] for the up-to-date progress in this direction.…”

Section: Introductionmentioning

confidence: 99%

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Complete regular dessins and skew-morphisms of cyclic groups

Feng¹,

Hu²,

Nedela³

et al. 2018

Preprint

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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,

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“…The next three properties of skew-morphisms are well known and were proved in [18,19,42]. For convenience, we include the proof of the third of them.…”

Section: Reciprocal Skew-morphismsmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Complete regular dessins and skew-morphisms of cyclic groups

Feng¹,

Hu²,

Nedela³

et al. 2018

Preprint

Self Cite

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“…Let ϕ be a skew morphism of a finite group A, let π be the power function of ϕ, and let k be the order of ϕ. Then µ = ϕ is a skew morphism of A if and only if the congruence x ≡ σ(a, ) (mod k) is soluble for every a ∈ A, in which case π µ (a) is the solution 15]). If ϕ is a skew morphism of a finite group A, then O −1 a = O a −1 for any a ∈ A, where O a and O a −1 denote the orbits of ϕ containing a and a −1 , respectively.…”

Section: Proposition 22 ([1]mentioning

confidence: 99%

Reflexible complete regular dessins and antibalanced skew morphisms of cyclic groups

Kwon

2020

Art Discrete Appl. Math.

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A skew morphism of a finite group A is a bijection ϕ on A fixing the identity element of A and for which there exists an integer-valued function π on A such that ϕ(ab) = ϕ(a)ϕ π(a) (b), for all a, b ∈ A. In addition, if ϕ −1 (a) = ϕ(a −1 ) −1 , for all a ∈ A, then ϕ is called antibalanced. In this paper we develop a general theory of antibalanced skew morphisms and establish a one-to-one correspondence between reciprocal pairs of antibalanced skew morphisms of the cyclic additive groups and isomorphism classes of reflexible regular dessins with complete bipartite underlying graphs. As an application, reflexible complete regular dessins are classified.

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“…We also refer to the mapping π : A → Z as a power function corresponding to f . Although the concept of a skew-morphism was introduced and investigated in the context of regular Cayley maps and hypermaps, see [3,5,8,9], Conder et al [4] pointed out that it appeared already in the context of factorisation of groups. Namely, let G be a finite group having a factorisation G = AB into subgroups A and B with B cyclic and A ∩ B = 1, and let b be a generator of B.…”

Section: Introductionmentioning

confidence: 99%