On Quasi-Essential Subgroups of Primary Abelian Groups

Abstract

All groups considered in this paper are abelian. A subgroup N of a group G is said to be a quasi-essential subgroup of G if G = 〈H, K〉 whenever H is an N-high subgroup of G and K is a pure subgroup of G containing N. We started the study of such subgroups in [5]; in particular, we characterized subsocles of a primary group which were both quasi-essential and centres of purity. In this paper we show that quasi-essential subsocles of a primary group are necessarily centres of purity answering thus in the affirmative a question raised in [5].We obtain the following theorem: A subsocle S of a p-group G is quasi-essential if and only if either S ⊂ G1or (pnG)[p] ⊃ S ⊃ (pn+1G)[p] for some non-negative integer n. The notation is that of [1]. If G is a group, thenwhere p is a prime integer.