Finite simple groups in which all maximal subgroups are π-closed. I

Abstract

Finite simple nonabelian groups G that are not π-closed for some set of primes π but have π-closed maximal subgroups (property (*) for (G, π)) are studied. We give a list L of finite simple groups that contains any group G with the above property (for some π). It is proved that 2 ∉ π for any pair (G, π) with property (*) (Theorem 1). In addition, we specify for any sporadic simple group G from L all sets of primes π such that the pair (G, π) has property (*) (Theorem 2). The proof uses the author’s results on the control of prime spectra of finite simple groups.