Automorphisms Of Finite P-groups Research Articles

A direct proof of Müller’s result on automorphisms of finite p-groups

A direct proof of Müller’s result on automorphisms of finite p-groups

Some remarks on subgroups, epimorphic images and p′-automorphisms of finite p-groups

Characterizations of some [Formula: see text]-groups are obtained: [Formula: see text]-groups of maximal class, metacyclic [Formula: see text]-groups. The critical subgroups for solvable groups are defined. For a number of known results new proofs are presented. For investigation of [Formula: see text]-automorphisms of [Formula: see text]-groups the minimal nonnilpotent groups are used essentially. Use of counting theorems allows us to simplify some proofs. A number of open problems is stated.

A note on automorphisms of finite p-groups

Let G be a finite non-cyclic p-group of order at least p^3. If G has an abelian maximal subgroup, or if G has an elementary abelian centre and is not strongly Frattinian, then the order of G divides the order the its automorphism group.

On a conjecture about automorphisms of finite p-groups

Let G be a nonabelian finite p-group. A longstanding conjecture asserts that G admits a noninner automorphism of order p. In this paper, we prove that if G satisfies one of the following conditions (1) \({\mathrm{rank}(G'\cap Z(G))\neq \mathrm{rank}(Z(G))}\) (2) \({\frac{Z_{2}(G)}{Z(G)}}\) is cyclic (3) C G (Z(Φ(G))) = Φ(G) and \({\frac{Z_{2}(G)\cap Z(\Phi(G))}{Z(G)} }\) is not elementary abelian of rank rs, where r = d(G) and s = rank (Z(G)), then G has a noninner central automorphism of order p which fixes Φ(G) elementwise.

On automorphisms of finite p-groups

Let G be a finite p-group such that xZ(G) ⊆ xG for all x ∈ G – Z(G), where xG denotes the conjugacy class of x in G. Then |G| divides |Aut(G)|, where Aut(G) is the automorphism group of G.

On a question about automorphisms of finite p-groups

This paper deals with an old problem: are there non-trivial finite p-groups which are isomorphic to their full automorphism group, besides the dihedral group of order 8? The answer (in the negative) is obtained in some special cases, including groups of class 2, powerful groups, groups with centre of prime order or an abelian subgroup of prime index, class-3 groups with cyclic centre, groups with coclass at most 3 and others.

Power automorphisms of finite p-groups

Power automorphisms of finite p-groups

Groups of automorphisms of finite p-groups

Thompson [1] showed that if p is an odd prime number, A is a p-group of operators of the finite group P in which the Frattini subgroup Φ(P) is elementary and central, and P/Φ(P) is a free ZpA-module, then Cp(A) covers CP/Φ(P)(A). Then he proposed the question of whether it is possible in this theorem to weaken the hypothesis that Φ(P) be elementary and central. In the work it is shown that this hypothesis may be replaced by a much weaker one; it is sufficient that P be met-Abelian and have nilpotence class <p. From examples it is seen that the restriction on the nilpotence class of P is essential. As a corollary we obtain the negative solution for all prime numbers p of Gaschutz's problem concerning the conjugacy of Hall p′-subgroups of Sylowizers of a p-subgroup of a solvable group [2].

Automorphisms of finite p-groups

Automorphisms of finite p-groups