Almost recognizability by spectrum for all simple groups in the list L

Establish that for each finite nonabelian simple group L from the list L, defined as follows (with q odd): (a) L_n(q) with 8 ≤ n ≤ 26 and n composite; (b) U_n(q) with 7 ≤ n ≤ 26; (c) S_{2n}(q) and O_{2n+1}(q) with 5 ≤ n ≤ 15 and n ≠ 8; (d) O^+_{2n}(q) with 5 ≤ n ≤ 18; and (e) O^-_{2n}(q) with 5 ≤ n ≤ 17 and n ≠ 8,16, the group L is almost recognizable by spectrum, meaning that for any finite group G with spectrum ω(G) equal to ω(L), one has L ≤ G ≤ Aut L (i.e., G is an almost simple group with socle isomorphic to L).

Background

The paper studies recognition of finite groups by spectrum, where the spectrum ω(G) denotes the set of element orders in G. Two groups are isospectral if their spectra coincide. The recognition problem is solved if the number h(G) of non-isomorphic finite groups isospectral to G is known and, if finite, these groups are described. For nonabelian simple groups, the property of being almost recognizable by spectrum—namely, that any isospectral group G satisfies L ≤ G ≤ Aut L—reduces the recognition problem to describing almost simple groups isospectral to L.

At the time of writing, the recognition by spectrum problem is solved for all nonabelian simple groups except those in the list L, comprising specified families of classical groups over odd q. The authors note a known conjecture asserting that all groups in L are almost recognizable by spectrum, and they show this conjecture is equivalent to ruling out cross-characteristic composition factors in isospectral groups (their Conjecture 1). This paper proves that equivalence for the subset of groups in L with disconnected prime graph, thereby resolving recognition for that subclass. The broader conjecture remains the central open direction for the full list L.