3-Generator Groups whose Elements Commute with Their Endomorphic Images Are Abelian

TL;DR

This paper proves that all 3-generator groups where each element commutes with its endomorphic images are necessarily abelian, establishing a key structural property of such groups and determining the minimal generators for non-abelian cases.

Contribution

It demonstrates that all 3-generator $E$-groups are abelian, providing a new classification result and establishing the minimal number of generators for non-abelian $E$-groups.

Findings

All 3-generator $E$-groups are abelian.

The minimal number of generators for non-abelian $E$-groups is four.

Provides a structural characterization of $E$-groups with few generators.

Abstract

A group in which every element commutes with its endomorphic images is called an $E$-group. Our main result is that all 3-generator $E$-groups are abelian. It follows that the minimal number of generators of a finitely generated non-abelian $E$-group is four.