A quantitative characterization of finite simple groups (Q1906605)

For a finite group \(G\) let \(i(G)= \{|G:M|\mid M\) maximal\} be the set of indexes of maximal subgroups. The author proves in this note: Theorem. Let \(A\) be an alternating group and \(B\) a nonabelian, finite, simple group with \(i(B)\subseteq i(A)\), then \(A\simeq B\). The proof of this result uses the classification of finite simple groups. The author conjectures that if \(M\), \(N\) are nonabelian simple groups with \(|N|\mid|M|\) and \(i(N) \subseteq i(M)\) then \(M\simeq N\) or \(N\simeq \text{PSL} (2,11)\) and \(M\simeq M_{11}\). Related results from \textit{D. Wang} and the author are [Acta Math. Sin. 35, No. 2, 273-278 (1992; Zbl 0794.20030), 37, No. 1, 108-115 (1994; Zbl 0826.20021), 37, No. 5, 601-606 (1994; Zbl 0813.20053) and Northeast. Math. J. 10, No. 1, 81-86 (1994; Zbl 0830.20043)].