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

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.

Key Words:

• 2-Engel groups
• Endomorphisms of groups
• Near-rings
• p-Groups

2000 Mathematics Subject Classification:

• 20D45
• 20E36

ACKNOWLEDGMENT

The authors thank the referee for his/her valuable comments for making the article shorter and clearer.

This work was supported partially by the Center of Excellence for Mathematics, University of Isfahan.

The research of the first author was in part supported by a grant from IPM (No. 87200118).

Notes

Communicated by M. R. Dixon.