Characterization of groups with finite periodic part
type: Item
instance of: scholarly article
title: Characterization of groups with finite periodic part
zbMATH Open document ID: 0577.20019
DOI: 10.1007/BF01979651
author: V. I. Senashov
author: V. P. Shunkov
published in: Algebra and Logic
publication date: 1983-01-01T00:00:00Z
review text: A point of a group G is an element x of finite order such that, (if \(x\neq 1)\) for every finite nontrivial subgroup K of G with \(x\in N_ G(K)\) the normalizer \(N_ G(K)\) has only finitely many finite subgroups containing x. The identity element is a point if and only if the set of torsion elements of G is finite. If \(x\in U\leq G\) is a point of G, it is also a point of U. The existence of points restricts the structure of the group substantially: for instance no group without involutions has an infinite locally finite subgroup containing a point (10.6). Main results: Theorem 1: Let G be a group without involutions and a, b two points of G such that \(<a,g^{-1}bg>\) is finite for almost all conjugates \(g^{-1}bg\). Then at least one of the following holds: (i) The set of torsion elements of G is finite, (ii) There is an infinite set M of finite subgroups of G such that for one of a,b (for a, say) and for all subgroups H in M the subgroup \(<H,a>\) is a finite Frobenius group with noninvariant factor \(\), the set of torsion elements in \(C_ G(H)\) is infinite and a is contained in only finitely many finite subgroups of \(<C_ G(H),a>\). - Theorem 2: A group G without involutions possesses only finitely many torsion elements if and only if there is a pair a, b of points such that, for almost all \(g^{-1}bg\), \(<a,g^{- 1}bg>\) is finite and whenever a or b is contained in a subgroup L with infinitely many torsion elements it is contained in infinitely many finite subgroups of L. - Several examples.
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 3922949
zbMATH Keywords: group without involutions
zbMATH Keywords: points
zbMATH Keywords: torsion elements
zbMATH Keywords: finite subgroups
reviewed by: Hermann Heineken
MaRDI profile type: Publication
cites work: Q4745979
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description: scientific article; zbMATH DE number 3922949