Skew-morphisms of cyclic p-groups (original) (raw)
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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.
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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}.
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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.
On Commuting Automorphisms of Some Finite p-Groups
Mathematical Notes, 2020
Let G be a group and Aut(G) be its automorphism 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). We find some cases when A(G) is a subgroup of Aut(G).
Mathematics and Statistics, 2014
In this paper, we have defined the concept of p-map and studied some properties of p-map. By using this map, we have shown that p(G) is a subgroup of G and S = {x : p(x) = e} is a right transversal (with identity) of p(G) in G which becomes group by using p-map and some more conditions. Finally, we have shown that G be an extension of p(G).
Some necessary and sufficient conditions on commuting automorphisms of some finite p-groups
Communications in Algebra, 2022
Let G be a finite non-abelian p-group of order greater than p 4 , p an odd prime, such that ZðX 1 ðHÞÞ is cyclic and Z(H) is elementary abelian, where H ¼ C G ðUðGÞÞ: We prove that the set AðGÞ ¼ fa 2 AutðGÞ j ½x, aðxÞ ¼ 1 for all x 2 Gg of all commuting automorphisms of G forms a subgroup of AutðGÞ if and only if X 1 ðHÞ is abelian. Also, we find the structure of AðGÞ ¼ Aut z ðGÞ for a finite 2-group G of almost maximal class with cyclic center Z(G), where Aut z ðGÞ denotes the set of all central automorphisms of G.