Projectively invariant subgroups of abelian \(p\)-groups
type: Item
instance of: scholarly article
title: Projectively invariant subgroups of abelian \(p\)-groups
zbMATH Open document ID: 1478.20037
DOI: 10.1134/S000143462105028X
published in: Mathematical Notes
publication date: 2021-07-08T00:00:00Z
review text: All groups are abelian. A subgroup \(C\) of a group \(G\) is said to be projectively invariant (briefly, a pi-subgroup) if \(\pi(C) \leq C\) for every projection \(\pi\) of \(G\). These groups have been studied by several authors, for example by \textit{C. K. Megibben} [Tamkang J. Math. 8, 177--182 (1977; Zbl 0421.20023)], \textit{J. Hausen} [Tamkang J. Math. 12, 215--218 (1981; Zbl 0493.16022)] and by author of the paper under this review [J. Math. Sci., New York 164, No. 1, 143--147 (2010; Zbl 1288.20076); translation from Fundam. Prikl. Mat. 14, No. 6, 211--218 (2008)]. Let \(C\) be a subgroup of a reduced \(p\)-group \(G\). With the subgroup \(C\), the author associates the sequence \(H(C)=(\alpha_0(C),\alpha_1(C),\ldots)\) where \(\alpha_k(C)=\min h_G(p^kC[p])\). The main result of the paper is the next theorem: if \(C\) is a projectively invariant subgroup of a reduced \(p\)-group \(G\) and \(\alpha_k(C)\) is integer number for any \(k\in \mathbb{N_0}\) then the subgroup \(C\) is fully invariant.
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 7369772
zbMATH Keywords: projection
zbMATH Keywords: projectively invariant subgroup
zbMATH Keywords: fully invariant subgroup
author: Andrey R. Chekhlov
reviewed by: Nikolay I. Kryuchkov
MaRDI profile type: Publication
cites work: Q3853826
cites work: Q3955537
cites work: On projective invariant subgroups of Abelian groups.
cites work: Q3144541
cites work: Q5609457
cites work: ON ENDOMORPHISM RINGS OF PRIMARY ABELIAN GROUPS, II
full work available at URL: s000143462105028x
OpenAlex ID: W3180563002
description: scientific article; zbMATH DE number 7369772