ON THE NUMBER OF CONJUGACY CLASSES IN FINITE $p$-GROUPS

Abstract

In this study of the behaviour of the number of conjugacy classes in finite p-groups using pro-p groups, the conjugacy growth function rn(G)+ max{r(G/N)|N◃0 G,|G:N|=n} is introduced. It is proved that there are no infinite pro-p groups of linear conjugacy growth (that is, there is no c such that rn(G)⩽ c log2 n for all n>1) and it is shown that many known pro-p groups are of exponential conjugacy growth (that is, there exists ε>0 such that rn(G)⩾nε for infinitely many values of n).