On the efficiency of direct powers of PGL(2, p)

A finite group G is said to be efficient if G has a presentation 〈 X | R 〉 where |R| = |X|+rank(M(G)) where M(G) is the Schur multiplier of G; see for example [14]. The efficiency of direct powers, G, of a finite group G has been studied over a number of years, see for example [5], [8] and [14]. In [6] the problem of proving that G is efficient for all n ∈ N, in the case of an imperfect group, is shown to reduce to proving that G is efficient for a finite number of values of n. Recently there has been interest in the semigroup efficiency of groups, see [2]. In particular the semigroup efficiency of direct powers of groups is considered in [1]. The purpose of this paper is to prove that G(p) is efficient for all odd primes p and all n ∈ N, where G(p) is given by the presentation