Antiautomorphisms and biantiautomorphisms of some finite abelian groups

TL;DR

This paper extends the concepts of antimorphisms and antiautomorphisms from cyclic groups to general finite abelian groups, providing bounds, formulas, and classifications related to these structures.

Contribution

It introduces new theoretical results on antiautomorphisms of finite abelian groups, including bounds, exact counts, and partial classifications.

Findings

Lower bound for antiautomorphisms of cyclic groups of odd order

Exact formula for linear antiautomorphisms of cyclic groups of odd order

Partial classification of finite abelian groups admitting antiautomorphisms

Abstract

We extend the concepts of antimorphism and antiautomorphism of the additive group of integers modulo $n$, given by Gaitanas Konstantinos, to abelian groups. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd order and give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. Finally, we give a partial classification of the finite abelian groups which admit antiautomorphisms and state some open questions.