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A group in which every element commutes with its endomorphic images is calledan $E$-group. If $p$ is a prime number, a $p$-group $G$ which is an $E$-groupis called a $pE$-group. Every abelian group is obviously an $E$-group. We provethat every 2-generator $E$-group is abelian and that all 3-generator $E$-groupsare nilpotent of class at most 2. It is also proved that every infinite3-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 anon-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 theclass of $E$-groups.

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