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20.79 (2022)

Open

Conjecture: Suppose that $G$ is a non-abelian simple group and $H$ is a finite group such that $\text{Cod}(G) = \text{Cod}(H)$ (see 20.78 for notation). Then $G \cong H$.

Contributor: G. Qian

Progress

Comment of 2025: The conjecture is proved for $G$ of type $\text{PSL}(2, q)$ (A. Bahri, Z. Akhlaghi, B. Khosravi, Bull. Austral. Math. Soc., 104, no. 2 (2021), 278–286); $\text{PSL}(3, q)$ or $\text{PSU}(3, q)$ (Y. Liu, Y. Yang, Results Math., 78, no. 1 (2023), article No. 7); a sporadic simple group (M. Dolorfino, L. Martin, Z. Slonim, Y. Sun, Y. Yang, Bull. Austral. Math. Soc., 109, no. 1 (2024), 57–66); an alternating group (M. Dolorfino, L. Martin, Z. Slonim, Y. Sun, Y. Yang, Bull. Austral. Math. Soc., 110, no. 1 (2024), 115–120); ${}^2F_4(q^2)$ (Y. Yang, J. Group Theory, 27, no. 1 (2024), 141–155); $\text{PSL}(n, q)$ or a simple exceptional group of Lie type (H. P. Tong Viet, Math. Nachr., 298, no. 4 (2025), 1356–1369).

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