Non-congruence presentations of finite simple groups

TL;DR

This paper proves that all non-abelian finite simple groups have non-congruence presentations and that those with a non-trivial Schur multiplier have smooth covers, confirming two conjectures in group theory.

Contribution

It establishes the existence of non-congruence presentations for all non-abelian finite simple groups and proves that groups with non-trivial Schur multipliers have smooth covers, advancing understanding in finite group theory.

Findings

Every non-abelian finite simple group admits a non-congruence presentation.

Finite simple groups with non-trivial Schur multipliers have smooth covers.

Proves two conjectures related to generators and covers of finite simple groups.

Abstract

We prove two results on some special generators of finite simple groups and use them to prove that every non-abelian finite simple group $S$ admits a non-congruence presentation (as conjectured in [CLT24]), and that if $S$ has a non-trivial Schur multiplier, then it admits a smooth cover (as conjectured in [CFLZ]).