On the efficiency of some p-groups

Abstract

Let p be a prime number. In this paper, we work on the efficiency of the p-groups G(1) and G(2) defined by the presentations,

P-G1 = < a, b, c; ab = bac, bc = cb, ac = ca, a(p alpha) = 1, b(p beta) = 1, c(p gamma) = 1 >

where alpha >= beta >= gamma >= 1 and

P-G2 = < a, b; ab = ba(1+p alpha-gamma), a(p alpha) = 1, b(p beta) = 1 >

where alpha >= 2 gamma, beta > gamma >= 1 and alpha+beta > 3. For example, if we let p = 2, then by [1], the groups defined by these presentations becomes 2-groups. It is known that these groups play an important role in the theory of groups of nilpotency class 2.