21.81 (2026)

(J. Wiegold). Let $\Gamma$ be a finite simple group and let $\mathcal{N}_n(\Gamma)$ denote the set of normal subgroups of the free group $F_n$ of rank $n$ whose quotient is isomorphic to $\Gamma$.

Conjecture: $\text{Aut}(F_n)$ acts transitively on $\mathcal{N}_n(\Gamma)$ for $n \geqslant 3$.

This is not true for $n = 2$ (B. H. Neumann, H. Neumann, Math. Nachr., 4 (1951), 106–125).