Alternating quotients 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 $\{{\gamma }_1,\dots ,{\gamma }_n\}\subseteq F\setminus H$ 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.