Determine the value d(M(G)) for non-abelian p-groups of order q = pnk of Nilpotency c

Abstract

In this paper we prove that if n, k and t be positive integer numbers such that t < k < n and G is a non abelian p-group of order pnk with derived subgroup of order pkt and nilpotency class c, then the minimal number of generators of G is at most p1 2 ((nt+kt−2)(2c−1)(nt−kt−1)+n. In particular, |M(G)| _ p1 2 (n(k+1)−2)(n(k−1)−1)+n, and the equality holds in this last bound if and only if n = 1 and G = H ×Z, where H is extra special p-group of order p3n and exponent p, and Z is an elementary abelian p-group.

Keywords

• schur multiplier, elementary abelian, p-group, extra special