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 G1 and G2 defined by the presentations,where α ≥ β > γ ≥ 1 andwhere α ≥ 2γ, β > γ ≥ 1 and α + β > 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. IntroductionLet G be a finitely presented group with a presentationThen the deficiency of this presentation is defined by |r| − |x|, and is denoted by def (P). Moreover, the group deficiency of a finitely presented group G is given by def G (G) = min{def (P) : P is a f inite group presentation f or G}.