On Commutativity and Finiteness in Groups

TL;DR

This paper explores the properties of groups generated by weakly commuting finite abelian groups, providing evidence supporting conjectures about their nilpotency and examining the concepts of weak permutability and commutativity.

Contribution

It introduces the notion of weak commutativity between groups and investigates the nilpotency of groups generated by such groups, extending previous concepts of weak permutability.

Findings

Groups generated by two weakly commuting finite abelian groups are likely nilpotent.

Supports conjectures on the nilpotency of these groups.

Provides theoretical insights into weak permutability and commutativity in group theory.

Abstract

The second author introduced notions of weak permutability and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite groups. Two groups H,K weakly commute provided there exists a bijection f: H -> K which fixes the identity element and such that h commutes with its image h^f for all h in H. The present paper gives support to conjectures about the nilpotency of groups generated by two weakly commuting finite abelian groups H,K.