Introduction

Our terminology and notation are mostly standard (see, for example, [1, 2]). We use the term “group” to mean “finite group.”

Let π be a set of primes. Denote by π_ the set of primes not in π. Given a natural n, we denote by π(n) the set of prime divisors of n. A natural number n with π(n) ⊆ π is called a π-number, and a group G such that π(G) ⊆ π is called a π-group. For a group G, the set π(G) = π(|G|) is the prime spectrum of G. A subgroup H of a group G is called a π-Hall subgroup if π(H) ⊆ π and π(|G : H|) ⊆ π’. Thus, if π consists of a single prime p then a π-Hall subgroup is exactly a Sylow p-subgroup. A Hall subgroup is a π-Hall subgroup for some set π of primes. A group G is prime spectrum minimal if π(H)≠ π(G) for every proper subgroup H of G.

We say that G is a group with Hall maximal subgroups if every maximal subgroup of G is a Hall subgroup. It is easy to see that every group with Hall maximal subgroups is prime spectrum minimal.

A group G is a group with complemented maximal subgroups if for every maximal subgroup M of G, there exists a subgroup H such that MH = G and M ∩ H = 1.

The study of groups with Hall maximal subgroups was started in 2006 by Levchuk and Likharev [3] and Tyutyanov [4], who established that a nonabelian simple group with complemented maximal subgroups is isomorphic to one of the groups PSL2(7) PSL3(2), PSL2(11) or PSL5(2). In all these groups, every maximal subgroup is a Hall subgroup. In 2008, Tikhonenko and Tyutyanov [5] showed that the nonabelian simple groups with Hall maximal subgroups are exhausted up to isomorphism by the groups PSL2(7), PSL2(11), and PSL5(2).