The group of automorphisms of an elementary-abelian-over-cyclic regular wreath product p-group

Abstract

Let W denote the regular wreath product finite group $C\wr E$ where C is a cyclic p-group and $E$ is an elementary abelian p-group. Let A denote the subgroup of $\text{Aut}(W)$ consisting of those automorphisms that act trivially on $W/B$, where B is the base group. We determine A by describing where each of its elements map a certain generating set for W. We find that A is as large as possible in a certain sense. We determine some information about the subgroup structure of A, and we prove that every class-preserving automorphism of W is an inner automorphism of W.

Keywords:

• Automorphism
• class-preserving automorphism; p-group
• wreath product

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D45
• 20D15