2007/08/28 by John R. Britnell, Mark Wildon, Britnell, John R. +1
Mathematics · #05D15 #20D60 #20E45 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:05D15 #msc:20D60 #msc:20E45
paper · pdf · doi:10.48550/arxiv.0708.3872
11 pages, revised and extended version
arxiv created 2008/10/25 · arxiv updated 2009/12/01
Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c ∈ C and d ∈ D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups.