A density theorem on the number of conjugacy classes in finite groups

Edward Arthur Bertram

Vol. 55 (1974), No. 2, 329–333

DOI: 10.2140/pjm.1974.55.329

Abstract

For each finite group G with k(G) conjugacy classes and order g, it is well known that g < 2^2k . On the other hand, all groups with a given small k(≦ 8) have been determined, and these studies, along with the result that if G is nilpotent then g < 2^k, strongly suggest that the bound can be significantly improved. We prove that for each c₂ < log 2, almost all integers g ≦ n, as n →∞, have the property that for each G of order g, k(G) > (log n)^c₂.

Mathematical Subject Classification 2000

Primary: 10H25

Secondary: 20D99

Milestones

Received: 22 May 1974

Revised: 26 November 1974

Published: 1 December 1974