Minimal number of generators and minimum order of a non-abelian group whose elements commute with their endomorphic images

Abstract

A group in which every element commutes with its endomorphic images is called an "E-group". If p is a prime number, a p-group G which is an E-group is called a "pE-group". Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups. Copyright © Taylor & Francis Group, LLC.