Skew-morphisms of cyclic p-groups (original) (raw)
Reciprocal skew morphisms of cyclic groups
2019
Reciprocal pairs of skew morphisms of cyclic groups are in one-to-one correspondence with isomorphism classes of regular dessins with complete bipartite underlying graphs. In this paper we determine all reciprocal pairs of skew morphisms of the cyclic groups provided that one of them is a group automorphism.
Decomposition of skew-morphisms of cyclic groups
Ars Mathematica Contemporanea, 2011
A skew-morphism of a group H is a permutation σ of its elements fixing the identity such that for every x, y ∈ H there exists an integer k such that σ(xy) = σ(x)σ k (y). It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups Z n : if n = n 1 n 2 such that (n 1 , n 2) = 1, and (n 1 , φ(n 2)) = (φ(n 1), n 2) = 1 (φ denotes Euler's function) then all skew-morphisms σ of Z n are obtained as σ = σ 1 × σ 2 , where σ i are skew-morphisms of Z ni , i = 1, 2. As a consequence we obtain the following result: All skew-morphisms of Z n are automorphisms of Z n if and only if n = 4 or (n, φ(n)) = 1.
Skew endomorphisms on some n-ary groups
2009
We characterize n-ary groups dened on a cyclic group and describe a group of their automorphisms induced by the skew operation. Finally, we consider splitting automorphisms.
Fixed Points of Automorphisms of Certain Non-Cyclic p-Groups and the Dihedral Group
Symmetry, 2018
Let G = Z p ⊕ Z p 2 , where p is a prime number. Suppose that d is a divisor of the order of G. In this paper, we find the number of automorphisms of G fixing d elements of G and denote it by θ(G, d). As a consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find the exact number of fixed-point-free automorphisms of the group Z p a ⊕ Z p b , where a and b are positive integers with a < b. Finally, we compute θ(D 2q , d), where D 2q is the dihedral group of order 2q, q is an odd prime, and d ∈ {1, q, 2q}.
On commuting automorphisms of finite p-groups
Mathematical Notes, 2021
Let G be a group. An automorphism α of G is called a commuting automorphism if α(x)x = xα(x) for all x ∈ G. The set of all commuting automorphisms of G is denoted by A(G). If A(G) forms a subgroup of Aut(G), then G is called an A(G)-group. Let G be a finite non-abelian p-group, where p is odd prime. We prove that if IA(G) = Inn(G) and |G/G ′ | = p 2. Then G is an A(G)-group. We also prove that if G is a finite non-abelian p-group of coclass 3, where p is an odd prime and G is strongly frattinian. Then G is an A(G)-group.
A note on p-groups of order ≤ p 4
Proceedings - Mathematical Sciences, 2009
In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p 4 , then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.
Journal of Pure and Applied Algebra, 2015
According to Li, Nicholson and Zan, a group G is said to be morphic if, for every pair N 1 , N 2 of normal subgroups, each of the conditions G/N 1 ∼ = N 2 and G/N 2 ∼ = N 1 implies the other. Finite, homocyclic p-groups are morphic, and so is the nonabelian group of order p 3 and exponent p, for p an odd prime. In this paper we show that these are the only examples of finite, morphic p-groups.