An endomorphism h of a group G is said to be strong whenever for every congruence θ on G , ( x,y ) ∈ θ implies ( h ( x ), h ( y )) ∈ θ for every x, y ∈ G. A group G is said to have the strong endomorphism kernel property if every congruence on G is the kernel of a strong endomorphism. In this note, we study the strong endomorphism kernel property in the class of Abelian groups. In particular, we show that a finite Abelian group has the strong endomorphism kernel property if and only if it is cyclic.

中文翻译：

---

具有强内同态核性质的有限阿贝尔群

每当对于 G 上的每个同余 θ，( x,y ) ∈ θ 暗示 ( h ( x ), h ( y )) ∈ θ 对于每个 x, y ∈ G 时，就说群 G 的自同态 h 是强的。如果 G 上的每个同余都是强自同态的核，则称 G 群具有强自同态核性质。在本笔记中，我们研究了阿贝尔群类中的强自同态核性质。特别地，我们证明了有限阿贝尔群具有强自同态核性质当且仅当它是循环的。