On the socles of characteristic subgroups of Abelian \(p\)-groups.
type: Item
instance of: scholarly article
title: On the socles of characteristic subgroups of Abelian \(p\)-groups.
zbMATH Open document ID: 1209.20047
published in: Journal of Algebra
publication date: 2010-06-08T00:00:00Z
review text: Recall that a \(p\)-group is said to be `socle regular' if for all fully invariant subgroups \(F\) of \(G\), there exists an ordinal \(\alpha\) such that \(F[p]=(p^\alpha G)[p]\). The socle regular groups were studied by the two authors in earlier work and it was shown that the class of socle-regular groups strictly contains the class of fully transitive groups. Thus this concept is closely related to the concept of (full-) transitivity introduced by Kaplansky. While the fully invariant subgroups have attracted much attention, the characteristic subgroups have been investigated only little. In this paper the authors introduce the stronger concept of `strongly socle-regular' meaning that for all characteristic subgroups \(C\) of \(G\), there exists an ordinal \(\alpha\) such that \(C[p]=(p^\alpha G)[p]\). Many nice properties of strongly socle-regular groups are investigated and the relation to the concepts of socle-regularity and (full-) transitivity is clarified. It is very interesting to see that some of the relationships mirror the relationship between fully transitive groups and transitive groups. Hence the new notions of socle-regular and strongly socle-regular groups can be seen as generalizations of Kaplansky's notions.
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 5718195
zbMATH Keywords: characteristic subgroups
zbMATH Keywords: Abelian \(p\)-groups
zbMATH Keywords: strongly socle-regular groups
zbMATH Keywords: fully transitive groups
zbMATH Keywords: fully invariant subgroups
author: Peter Danchev
author: B. Goldsmith
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.12.005
OpenAlex ID: W2062949860
cites work: Q4342195
cites work: ON ENDOMORPHISM RINGS OF PRIMARY ABELIAN GROUPS
cites work: THE INDEPENDENCE OF KAPLANSKY'S NOTIONS OF TRANSITIVITY AND FULL TRANSITIVITY
cites work: On the socles of fully invariant subgroups of Abelian \(p\)-groups.
cites work: Transitive and fully transitive groups
cites work: Q5609457
cites work: Two classes of rings generated by their units
cites work: On Transitive and Fully Transitive Primary Groups
cites work: Q5829819
cites work: Fully transitive $p$-groups with finite first Ulm subgroup
cites work: Q3734710
cites work: Cancellation in Direct Sums of Groups
DOI: 10.1016/J.JALGEBRA.2009.12.005
description: scientific article; zbMATH DE number 5718195