Classification of finite simple families with bounded-length mixed identities

Characterize all cofinite subfamilies of pairwise nonisomorphic nonabelian finite simple groups satisfying mixed identities of uniformly bounded length, specifically determining whether only alternating groups, groups of Lie type over fields of bounded size, and the listed symplectic, odd-dimensional orthogonal, F_4, and G_2 families occur.

Background

The conjecture proposes a complete description of families of pairwise nonisomorphic nonabelian finite simple groups that admit mixed identities whose lengths are uniformly bounded. It includes both a necessity statement, up to a cofinite subfamily, and a converse asserting bounded-length mixed identities for the listed families. The paper proves this conjecture except possibly for families containing infinite sequences of even-degree orthogonal groups with both rank and field size tending to infinity.

References

We have a conjectural description of those finite simple groups satisfying mixed identities of bounded length. If $\mathcal{F}$ is a family of pairwise nonisomorphic nonabelian finite simple groups satisfying mixed identities of bounded length, then $\mathcal{F}$ contains a cofinite subfamily consisting of groups of the following types: the alternating groups $\Alt_n$; groups of Lie type over fields of bounded size; $C_m(q)=\PSp_{2m}(q)$; $B_m(q)'=\POmega_{2m+1}(q)$; $F_4(q)$; $G_2(q)$. Conversely, groups of types (i), (ii) or (iii) do satisfy mixed identities of bounded length.

— Mixed identities for simple locally finite groups  (2608.14537 - Bradford et al., 14 Aug 2026) in Introduction, Conjecture labeled \ref{IntroFSGMainConj}