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Finite groups with arithmetic restrictions on maximal subgroups. (English. Russian original) Zbl 1323.20024

Algebra Logic 54, No. 1, 65-69 (2015); translation from Algebra Logika 54, No. 1, 95-102 (2015).
From the introduction: 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 \(\pi\)-properties (properties of a group associated with some set \(\pi\) 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.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D30 Series and lattices of subgroups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D25 Special subgroups (Frattini, Fitting, etc.)
20E28 Maximal subgroups
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