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 $r_n(G)= \max\{r(G/N)\,{|}\,N\triangleleft_o G,|G\,{:}\,N|=n\}$ 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 $r_n(G)\le c\log_2 n$ for all $n>1$) and it is shown that many known pro-$p$ groups are of exponential conjugacy growth (that is, there exists $\epsilon\,{>}\,0$ such that $r_n(G)\,{\ge}\,n^\epsilon$ for infinitely many values of $n$).