Remarks on Proficient groups

TL;DR

This paper investigates proficient presentations in finite and profinite groups, demonstrating that many perfect groups, including infinitely many alternating and symmetric groups, possess such efficient presentations.

Contribution

It extends the concept of proficient presentations from finite to profinite groups and shows many perfect groups have these efficient presentations, including infinite families.

Findings

Many perfect groups have proficient presentations

Infinitely many alternating and symmetric groups are proficient

Double covers of these groups also have proficient presentations

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