21.47 (2026)

(Well-known problem). Does there exist a finitely presented group $G$ such that $G \cong G \times H$ for some non-trivial group $H$?

The first finitely generated example was constructed in (J. M. Tyrer Jones, J. Austral. Math. Soc., 17 (1974), 174–196). A finitely presented group that surjects onto its own direct square was constructed in (G. Baumslag, C. F. Miller, III, Bull. London Math. Soc., 20, no. 3 (1988), 239–244).