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1 results

  • Weak Cayley tables and generalized centralizer rings of finite groups

  • STEPHEN P. HUMPHRIES, EMMA L. RODE
  • Journal:
    Mathematical Proceedings of the Cambridge Philosophical Society / Volume 153 / Issue 2 / September 2012
    Published online by Cambridge University Press:
    14 August 2012, pp. 281-318
    Print publication:
    September 2012

  • 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.