Weak Cayley tables and generalized centralizer rings of finite groups

Abstract

AbstractFor a finite groupGwe study certain rings(k)Gcalledk-S-rings, one for eachk≥ 1, where(1)Gis the centraliser ringZ(ℂG) ofG. These rings have the property that(k+1)Gdetermines(k)Gfor allk≥ 1. We study the relationship of(2)Gwith the weak Cayley table ofG. We show that(2)Gand the weak Cayley table together determine the sizes of the derived factors ofG(noting that a result of Mattarei shows that(1)G=Z(ℂG) does not). We also show that(4)GdeterminesGfor any groupGwith 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.