Endomorphism kernel property for finite groups

Abstract

A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $\theta$ on $G$ is the kernel of an endomorphism on $G$. In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

Keywords

• endomorphism kernel property
• nilpotent group
• $p$-group