Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

Daniel López-Aguayo,Servando López Aguayo

doi:10.3390/sym12020294

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.

Highlights

In this paper we introduce the concept of biantiautomorphism which is a bijective antimorphism in the sense of [1]
The main result of this paper is Theorem 2 which gives a partial classification of the finite abelian groups which admit antiautomorphisms
We give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order

Summary

Introduction

In this paper we introduce the concept of biantiautomorphism which is a bijective antimorphism in the sense of [1]. We give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. To obtain p2 − 2p antiautomorphisms, it suffices to consider all the translation maps of φ a ; that is, ψa,b (t) := φ a (t) + b where b ∈ Z p. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd prime power order. Let (m, n) denote the greatest common divisor of the integers m and n It suffices to compute the cardinality of the set W = { a ∈ [2, pα − 1] : ( a, pα ) = ( a − 1, pα ) = 1}.

Antiautomorphisms

Findings

Biantiautomorphisms