Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem

Details Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem

2007-08-28

arxiv.org/abs/0708.3872v2

Mathematics

Group Theory

11 pages, revised and extended version

Scientific paper

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in 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.

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