On automorphism groups and endomorphism rings of abelian 𝑝-groups

Abstract

Let A A be a noncyclic abelian p p -group where p ⩾ 5 p \geqslant 5 , and let p ∞ A {p^\infty }A be the maximal divisible subgroup of A A . It is shown that A / p ∞ A A/{p^\infty }A is bounded and nonzero if and only if the automorphism group of A A contains a minimal noncentral normal subgroup. This leads to the following connection between the ideal structure of certain rings and the normal structure of their groups of units: if the noncommutative ring R R is isomorphic to the full ring of endomorphisms of an abelian p p -group, p ⩾ 5 p \geqslant 5 , then R R contains minimal twosided ideals if and only if the group of units of R R contains minimal noncentral normal subgroups.