On the number of conjugacy classes in a finite p-group (Q5906251)

Let p be a prime, G be a finite group of order \(p^ m=p^{2n+e}\) with \(n\geq 0\) and \(e=1\) or 0, and let r(G) be the number of conjugacy classes of G. P. Hall proved that \(r(G)=(p^ 2-1)n+p^ e+k(p^ 2-1)(p-1)\), for some integer \(k\geq 0\). In this paper new inequalities of this type are proved. For example, Theorem 1. There exist non-negative integers \(k_ 1\) and \(k_ 2\) such that \[ 1)\quad p\cdot r(G)=(p^ 2-1)(| Z(G)| +n+e+p-2)+p^{1-e}+k_ 1(p^ 2-1)(p-1); \] \[ 2)\quad p\cdot r(G)=(p^ 2-1)(| G:G'| /p+n+e+p-2)+p^{1-e}+k_ 2(p^ 2-1)(p- 1). \] Corollaries 2 and 6 give formulas for r(G) when \(n\leq p+1\) and G is of maximal class, resp.