The group of automorphisms of an elementary-abelian-over-cyclic regular wreath product \(p\)-group. (English) Zbl 1472.20047
Summary: 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 \(\operatorname{Aut}(W)\) 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 \(W\).
MSC:
| 20D45 | Automorphisms of abstract finite groups |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20E22 | Extensions, wreath products, and other compositions of groups |
| 20F28 | Automorphism groups of groups |
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