Finite Groups with Arithmetic Restrictions on Maximal Subgroups

Abstract

In group theory, properties of a group that are defined by its numerical parameters are conventionally referred to as arithmetic. Among these are the order of a group and the set of its prime divisors, orders of elements, orders of subgroups, degrees of irreducible representations, various π-properties (properties of a group associated with some set π of primes, for instance, theorems of Sylow type), and so on. The term the normal structure of a group characterizes group invariants such as sets of composition and chief factors with due regard for the specific features of the action of the group on these factors. It is well known that there is a strong mutual influence between arithmetic properties and the normal structure of a finite group. In the present paper, we present some results on the normal structure of finite groups with arithmetic restrictions on maximal subgroups. In what follows, the term a ‘group’ is used to refer to a finite group. The spectrum of a group G is the set ω(G) of all element orders of G. The set of all primes occurring in the spectrum of G is called the prime spectrum of G and is denoted by π(G). The spectrum ω(G) defines the Gruenberg–Kegel graph (or prime graph) Γ(G) of G such that vertices in Γ(G) are primes of π(G), and two vertices r and s are adjacent if the number rs belongs to the set ω(G). A subgroup H of a group G is called a Hall subgroup if its order |H| and index |G : H| are coprime. We refer to G as a group with maximal Hall subgroups if its maximal subgroups each is ∗Supported by RFBR (project No. 13-01-00469), by Dmitry Zimin’s Dynasty Foundation and the Program for State Aid of Leading RF Universities (Agreement No. 02.A03.21.0006 between the Ministry of Education and Science of the Russian Federation and the Ural Federal University, 27.08.2013), and by the Complex Research Program of UrO RAN (project No. 15-16-1-5).