Geodesic
On Quasi-Essential Subgroups of Primary Abelian Groups
Canadian journal of mathematics, Tome 22 (1970) no. 6, pp. 1176-1184

Voir la notice de l'article provenant de la source Cambridge University Press

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 ⊂ G 1or (pnG)[p] ⊃ S ⊃ (p n+1 G)[p] for some non-negative integer n. The notation is that of [1]. If G is a group, then where p is a prime integer. 
Benabdallah, Khalid; Irwin, John M. On Quasi-Essential Subgroups of Primary Abelian Groups. Canadian journal of mathematics, Tome 22 (1970) no. 6, pp. 1176-1184. doi: 10.4153/CJM-1970-135-9
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