Upper Bounds for the Number of Conjugacy Classes of a Finite Group
Authors: Liebeck, M.W.; Neumann, L.P.M.
Source: Journal of Algebra, Volume 198, Number 2, 1 December 1997 , pp. 538-562(25)
Publisher: Academic Press
Abstract:For a finite group G, let k(G) denote the number of conjugacy classes of G. We prove that a simple group of Lie type of untwisted rank l over the field of q elements has at most (6q)l conjugacy classes. Using this estimate we show that for completely reducible subgroups G of GL(n, q) we have k(G) < q10n, confirming a conjecture of Kovacs and Robinson. For finite groups G with F*(G) a p-group we prove that k(G) < (cp)a where pa is the order of a Sylow p-subgroup of G and c is a constant. For groups with Op(G) = 1 we obtain that k(G) < |G|p\'. This latter result confirms a conjecture of Iranzo, Navarro, and Monasor. We also improve various earlier results concerning conjugacy classes of permutation groups and linear groups. As a by-product we show that any finite group G has a soluble subgroup S and a nilpotent subgroup N such that k(G) < |S| and k(G) < |N|3. Copyright 1997 Academic Press.
Document Type: Research Article
Affiliations: Department of Mathematics, Imperial College, London, SW7 2BZ, United Kingdom
Publication date: 1 December 1997