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Weak Cayley tables and generalized centralizer rings of finite groups

Published online by Cambridge University Press:  14 August 2012

STEPHEN P. HUMPHRIES
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA e-mail: steve@math.byu.edu; et@math.byu.edu
EMMA L. RODE
Affiliation:
Department of Mathematics, Brigham Young University, Provo, UT 84602, USA e-mail: steve@math.byu.edu; et@math.byu.edu

Abstract

For a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2012

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