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A one-sided Auslander--Lyndon theorem
We prove that if $F$ is a noncyclic free group, $R\triangleleft F$ and $S\le F$, then the inclusion $R'\le S'$ implies $R\le S$. Thus the normality hypothesis on the second subgroup can be omitted from the Auslander--Lyndon theorem. No restrictions are imposed on ranks or indices.
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