This paper deals with the determination of the automorphism group of the metacyclic p-groups, P (p,m), given by the presentation P (p,m) = 〈x, y|xpm = 1, y = 1, yxy−1 = xp+1〉 (1) where p is an odd prime number and m > 1. We will show that Aut(P ) has a unique Sylow p-subgroup, Sp, and that in fact

Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

Let FG be a group algebra of a finite non-abelian pgroup G and F a field of characteristic p. In this paper we give all minimal non-abelian p-groups and minimal non-metacyclic p-groups whose group algebras FG possess a filtered multiplicative F -basis.

In this paper we find the complete structure for the automorphism groups of metacyclic minimal nonabelian 2-groups. This, together with [6, 7], gives the complete answer to the Question 15 from [5] (respectively Question 20 from [4]) in the case of metacyclic groups. We also correct some inaccuracies and extend the results from [13].

Special covers are metacyclic covers of the projective line, with Galois group Z/p ⋊ Z/m, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of P with three branch points. Our results provide a classification of all special covers in terms of certain lifting data in characteristic p.

Given a finite metacyclic group G, a central extension F having the projective lifting property over all fields is constructed. This extension and its group rings are used to investigate the faithful irreducible projective representations of G and the fields over which they can be realized. A full description of the finite metacyclic groups having central simple twisted group rings over fields ...

Suppose that a finite solvable group G acts faithfully, irreducibly and quasi-primitively on vector space V, is not metacyclic. Then always has regular orbit V except for few “small” cases.

let $g$ be a finite group‎. ‎a subset $x$ of $g$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $x$ do not commute‎. ‎in this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.