2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs

TL;DR

This paper classifies specific 2-groups that factor as products of cyclic groups with an automorphism of order 2, providing explicit classifications, presentations, and automorphism groups for these groups.

Contribution

It extends the classification of 2-groups with particular factorization properties, including non-metacyclic groups, and simplifies their presentations and automorphism group descriptions.

Findings

Exactly three non-metacyclic groups for each e>2 with |A|=|B|=2^e

One such group for e=2

Explicit automorphism groups for classified groups

Abstract

We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B, transposed by an automorphism of order 2. The case where G is metacyclic having been dealt with elsewhere, we show that for each e>2 there are exactly three such non-metacyclic groups G with $|A|=|B|=2^e$, and for e=2 there is one. These groups appear in a classification by Berkovich and Janko of 2-groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.