# Alternating quotients of free groups

### Henry Wilton

University of Cambridge, Great Britain

## 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 $\{\gamma_1,\ldots,\gamma_n\}\subseteq F\smallsetminus H$ be a finite subset. Then there is a surjection $f$ from $F$ to a finite alternating group such that $f(\gamma_i)\notin f(H)$ for any $i$. The techniques of this paper can also provide symmetric quotients.

## Cite this article

Henry Wilton, Alternating quotients of free groups. Enseign. Math. 58 (2012), no. 1/2, pp. 49–60

DOI 10.4171/LEM/58-1-2