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On Quasi-Essential Subgroups of Primary Abelian Groups

Published online by Cambridge University Press:  20 November 2018

Khalid Benabdallah  and
John M. Irwin
Khalid Benabdallah
Affiliation:
Université de Montréal, Montréal, Québec;
John M. Irwin
Affiliation:
Université de Montréal, Montréal, Québec; Wayne State University. Detroit, Michigan
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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, then

where p is a prime integer.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1176 - 1184
Copyright
Copyright © Canadian Mathematical Society 1970

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