Upper Bounds for the Number of Conjugacy Classes of a Finite Group Original Research Article

For a finite groupG, letk(G) denote the number of conjugacy classes ofG. We prove that a simple group of Lie type of untwisted ranklover the field ofqelements has at most (6q)lconjugacy classes. Using this estimate we show that for completely reducible subgroupsGofGL(n, q) we havek(G) ≤ q10n, confirming a conjecture of Kovács and Robinson. For finite groupsGwithF*(G) ap-group we prove thatk(G) ≤ (cp)awherepais the order of a Sylowp-subgroup ofGandcis a constant. For groups withOp(G) = 1we obtain thatk(G) ≤ Gp′. 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 groupGhas a soluble subgroupSand a nilpotent subgroupNsuch thatk(G) ≤ S andk(G) ≤ N3.