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On Deficient Squares Groups and Fully‐Independent Subsets

On Deficient Squares Groups and Fully‐Independent Subsets ON DEFICIENT SQUARES GROUPS AND FULLY- INDEPENDENT SUBSETS MARCEL HERZOG AND CARLO M. SCOPPOLA 1. Introduction In 1976, B. H. Neumann [4] characterized the central-by-finite groups by the following property: the group G is central-by-finite if and only if it does not contain an infinite independent subset, where a subset U of G is called independent if uv = vu for u,veUimplies u = v. More recently, P. Longobardi, M. Maj and the first author [2] considered a more general class of groups, the deficient squares groups. A group is said to satisfy the deficient squares property if there exists an integer m such that 2 2 \M*\ < |A/j for all subsets M of G of size m, where M = {mn \m,ne M). They showed that a group G satisfies the deficient squares property if and only if one of the following holds: G is central-by-finite, |{g |ge(j}| is finite, or G is nearly dihedral, where by nearly dihedral we mean a group G with an abelian normal subgroup H of finite index, on which each element of G acts by conjugation either as the identity automorphism or as the inverting automorphism. The aim of this paper http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Deficient Squares Groups and Fully‐Independent Subsets

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References (4)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/27.1.65
Publisher site
See Article on Publisher Site

Abstract

ON DEFICIENT SQUARES GROUPS AND FULLY- INDEPENDENT SUBSETS MARCEL HERZOG AND CARLO M. SCOPPOLA 1. Introduction In 1976, B. H. Neumann [4] characterized the central-by-finite groups by the following property: the group G is central-by-finite if and only if it does not contain an infinite independent subset, where a subset U of G is called independent if uv = vu for u,veUimplies u = v. More recently, P. Longobardi, M. Maj and the first author [2] considered a more general class of groups, the deficient squares groups. A group is said to satisfy the deficient squares property if there exists an integer m such that 2 2 \M*\ < |A/j for all subsets M of G of size m, where M = {mn \m,ne M). They showed that a group G satisfies the deficient squares property if and only if one of the following holds: G is central-by-finite, |{g |ge(j}| is finite, or G is nearly dihedral, where by nearly dihedral we mean a group G with an abelian normal subgroup H of finite index, on which each element of G acts by conjugation either as the identity automorphism or as the inverting automorphism. The aim of this paper

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jan 1, 1995

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