About the autotopisms of abelian groups

TL;DR

This paper characterizes the autotopism group of any abelian group as a semidirect product of its automorphism group and G^2, and details its subgroup structure for finite cyclic groups.

Contribution

It provides a general description of autotopism groups for abelian groups and explicitly analyzes the subgroup structure for finite cyclic cases.

Findings

Autotopism group is a semidirect product of Aut(G) and G^2.

Explicit subgroup structure for finite cyclic groups.

General framework applicable to all abelian groups.

Abstract

We describe the autotopism group Atp(G) of any abelian group G as being a semidirect product of its automorphism group Aut(G) and G^2. We then provide the subgroup structure of Atp(G) when G is a finite cyclic group.