The {ie210-01}-property in finite simple groups

Abstract

Let π be some set of primes. A finite group is said to possess the D π -property if all of its maximal π-subgroups are conjugate. It is not hard to show that this property is equivalent to satisfaction of the complete analog of Sylow's theorem for Hall π-subgroups of a group. In the paper, we bring to a close an arithmetic description of finite simple groups with the D π -property, for any set π of primes. Previously, it was proved that a finite group possesses the D π -property iff each composition factor of the group has this property. Therefore, the results obtained mean in fact that the question of whether a given group enjoys the D π -property becomes purely arithmetic.Hall π-subgroup, then G is said to possess the D π -property [1]. A group with the E π -(C π -, D π -) property is also referred to as an E π -(C π -, D π -) group, respectively.Hall subgroups and groups with the D π -property were studied by many well-known authors (see, e.g., ). A classical result in this direction is a celebrated theorem by Hall, which was proved in [2,3], and independently, in [4]. The Hall theorem, we recall, says that a finite group is soluble iff it has the D π -property, for any set π of primes.Let π be fixed. Then the class of groups with the D π -property may turn out to be substantially wider than the class of all soluble groups, since, say, every π-or π -group possesses this property. Thus, it seems natural to pose the following: Problem 1. For every set π, describe all groups with the D π -property. The present paper brings to a close a series of works [28][29][30][31][32][33] devoted to this problem and contains a result which solves it out.The Hall theorem demonstrates that there exists a close connection between the composition structure of a finite group and the existence of the D π -property in that group, for all π. It turns out that this connection obtains even if the set π is fixed. It is not hard to show that every factor group of a group with the D π -property also has this property. Below are some problems which have been intensively studied in the mathematical literature starting in the 1950s (see [1, 5-13, 19, 20, 22]). Problem 2. Will an extension of a groupA by a group B possess the D π -property if A and B have this property? Problem 3. Will a normal subgroup of a group with the D π -property possess this property? In the general setting, Problem 2 was formulated in [13; 34, Question 3.62], and Problem 3 in [22; 34, Question 13.33]. Ph. Hall brought out a positive solution to Problem 2 for the case where a Hall π-subgroup is nilpotent in A and is soluble in B (see [1, Thm. D5]). Shemetkov in [10] solved this problem for the case where Sylow p-subgroups of A are cyclic for all p ∈ π, and obtained a series of other important results in this direction (see [11,12]). In [19] and [20], first attempts were made to use the theory of simple groups in dealing with Problem 2. Although a positive solution was obtained for several particular cases, the general approach to Problems 2 and 3 was outlined only in ...

Теоремы Силовского Типа

“…Из [74], [75], [77]- [79], [98]. Тем самым проблема 6.8 полностью решена (mod CFSG), и ее решение содержит следующая теорема.…”