The automorphism group of $F_2$(the free group generated by two elements) is finitely generated. Especially, it is generated by Nielsen transformations. If we consider all homomorphisms from $F_2$ to itself, not only these are invertible, we can get an endomorphism monoid of $F_2$. Is it also finitely generated? I tried to find some reference about it, but without result. Should I try to read textbooks about Nielsen transformation and try to generalize it?
2026-03-27 21:20:43.1774646443
Endomorphism monoid of free group
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No. Let me do it for the free group $F_d$ for arbitrary $d\ge 1$ ($d=1$ was done in a comment by user freakish).
Each endomorphism induces an endomorphism of the abelianization $\mathbf{Z}^d$, and composing with determinant yields a surjective homomorphism $\mathrm{End}(F_d)\to(\mathbf{Z},\times)$. Since the latter is not finitely generated as monoid, it follows that $\mathrm{End}(F_d)$ is not finitely generated either.