Alternating quotients of free groups

TL;DR

This paper extends Marshall Hall's Theorem by demonstrating that free groups can be mapped onto finite alternating groups in a way that separates certain elements from subgroups, strengthening residual properties.

Contribution

It introduces the concept of locally extended residually alternating free groups and provides methods to obtain symmetric quotients, advancing the understanding of free group quotients.

Findings

Free groups of rank at least two can be mapped onto finite alternating groups.

The techniques can produce symmetric quotients.

Strengthens the residual properties of free groups.

Abstract

We strengthen Marshall Hall's Theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let {g_1,...,g_n} be a finite subset of F-H. Then there is a surjection f from F to a finite alternating group such that f(g_i) is not in f(H) for any i. The techniques of this paper can also provide symmetric quotients.