Remarks on proficient groups

Abstract

If a finite group G has a presentation with d generators and r relations, it is well known that r − d is at least the rank of the Schur multiplier of G; a presentation is called efficient if equality holds. There is an analogous definition for proficient profinite presentations. We show that many perfect groups have proficient presentations. Moreover, we prove that infinitely many alternating groups, symmetric groups and their double covers have proficient presentations.

Presentations of finite simple groups: a computational approach

“…Section 3.5 contains explicit presentations of S n for all n ≥ 50. For general n it would be desirable to have even fewer relations than in Theorem C, with the goal of approaching efficiency for alternating groups (compare [GKKL3], where we use profinite presentations for a similar purpose).…”