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21.14 (2026)

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Suppose $\alpha$ is an endomorphism of a group $G$ such that for every group $H$ and every homomorphism $f : G \to H$, there exists an endomorphism $\beta_f$ of $H$ such that $\beta_f f = f\alpha$. Must $\alpha$ then be either an inner automorphism of $G$ or the trivial endomorphism? This is Question 5 in (G. M. Bergman, Publ. Matem., 56 (2012), 91–126).

Contributor: G. M. Bergman

Progress

*Yes, it must (F. Fournier-Facio, Preprint, 2026, https://arxiv.org/abs/2604.05728).

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