Upper bounds for the number of conjugacy classes of a finite group

[摘要] 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) less than or equal to q(10n), confirming a conjecture of Kovacs and Robinson. For finite groups G with F*(G) a p-group we prove that k(G) less than or equal to (cp)(a) where p(a) is the order of a Sylow p-subgroup of G and c is a constant. For groups with O-p(G) = 1 we obtain that k(G) less than or equal to \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) less than or equal to \S\ and k(G) less than or equal to \N\(3). (C) 1997 Academic Press.

[发布日期] 1997-12-15