The automorphism group of a split metacyclic p-group

Abstract

This paper finds the order, structure and presentation for the automorphism group of a split metacyclic p-group, where p is odd.

On the structure of the automorphism group of a minimal nonabelian p-group (metacyclic case)

“…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. In Section 2 we generalize the results from [13] and we specify a method of finding relations in an automorphism group, that we will use in the next Sections.…”

“…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].…”

Automorphism groups of semidirect products

“…That is, G = H K is a semidirect product of H by K. For any θ ∈ Aut G, let θ(h) = α(h)γ(h) and θ(k) = β(k)δ(k) for all h ∈ H, k ∈ K, where α : H → H, β : K → H, γ : H → K and δ : K → K are maps uniquely determined by θ. For convenience we shall write θ = α β γ δ , a 2 × 2 matrix of maps which satisfy all the conditions in Lemma 2.1 in [1].…”

“…We shall use the notation in [1]. There h k = khk −1 for h ∈ H, k ∈ K, and Aut C K (H) (K) = {δ ∈ Aut K|k −1 δ(k) ∈ C K (H), ∀ k ∈ K}.…”

“…Clearly this set is a group under the usual additive operation, namely (λ+λ )(k) = λ(k)λ (k) for λ, λ ∈ CrossHom(K, Z(H)). Now, by Lemma 2.1 in [1] it is easy to verify that…”