Minimal Number of Generators and Minimum Order of a Non-Abelian Group whose Elements Commute with Their Endomorphic Images

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 $p^8$ for any odd prime number $p$ and this order is $2^7$ for $p=2$. Some of these results are proved for a class wider than the class of $E$-groups.