The free metabelian group of exponent 𝑝²

Abstract

Introduction. For n_2 and p a prime, let Mn,,2 denote the free metabelian group of exponent p2 on n generators. The precise nilpotency class of M.,4 (n> 2) was established in Gupta and Tobin [2 ], where it was also shown that I M2,41 = 2 10 and M33,41 = 23. In a recent paper [1], Bachmuth and Mochizuki have shown that the class of M2,,2 (p odd) is precisely 2(p2-p). In another paper (to appear) Bachmuth, Heilbronn and Mochizuki have shown that for 3<k<p+1, the class of iIk,p2 is 2 (p2-p) or 2(p2-p)+1. In this paper we complete the discussion for the case k _ p +2.