D. O. Revin

Pronormality of Hall subgroups in finite simple groups

We prove that the Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of the Kourovka Notebook.

On the pronormality of hall subgroups

Fix a set of primes π. A finite group is said to satisfy Cπ or, in other words, to be a Cπ-group, if it possesses exactly one class of conjugate π-Hall subgroups. The pronormality of π-Hall subgroups in Cπ-groups is proven, or, equivalently, we show that Cπ is inherited by overgroups of π-Hall subgroups. Thus an affirmative solution is obtained to Problem 17.44(a) from The Kourovka Notebook. We also provide some example demonstrating that Hall subgroups in finite groups are not pronormal in general.

On the pronormality of subgroups of odd index in finite simple groups

We prove the pronormality of subgroups of finite index for many classes of simple groups.

The existence of pronormal π-Hall subgroups in E π -groups

A subgroup H of a group G is called pronormal, if the subgroups H and Hg are conjugate in 〈H, Hg〉 for every g ∈ G. It is proven that if a finite group G possesses a π-Hall subgroup for a set of primes π, then its every normal subgroup (in particular, G itself) has a π-Hall subgroup pronormal in G.

On the class of groups with pronormal hall π-subgroups

Given a set π of prime numbers, we define the class of all finite groups in which Hall π-subgroups exist and are pronormal by analogy with the Hall classes , , and . We study whether is closed under the main class-theoretic closure operations. In particular, we establish that is a saturated formation.

On Baer-Suzuki π-theorems

Given a set π of primes, say that the Baer-Suzuki π-theorem holds for a finite group G if only an element of Oπ(G) can, together with each conjugate element, generate a π-subgroup. We find a sufficient condition for the Baer-Suzuki π-theorem to hold for a finite group in terms of nonabelian composition factors. We show also that in case 2 ∉ π the Baer-Suzuki π-theorem holds for every finite group.

On a relation between the Sylow and Baer-Suzuki theorems

Given a set π of primes, say that a finite group G satisfies the Sylow π-theorem if every two maximal π-subgroups of G are conjugate; equivalently, the full analog of the Sylow theorem holds for π-subgroups. Say also that a finite group G satisfies the Baer-Suzuki π-theorem if every conjugacy class of G every pair of whose elements generate a π-subgroup itself generates a π-subgroup. In this article we prove, using the classification of finite simple groups, that if a finite group satisfies the Sylow π-theorem then it satisfies the Baer-Suzuki π-theorem as well.

Confirmation for Wielandt's conjecture ∗

Abstract Let π be a set of primes. By H. Wielandts definition, Sylow π-theorem holds for a finite group G if all maximal π -subgroups of G are conjugate. In the paper, the following statement is proven. Assume that π is a union of disjoint subsets σ and τ and a finite group G possesses a π -Hall subgroup which is a direct product of a σ -subgroup and a τ -subgroup. Furthermore, assume that both the Sylow σ -theorem and τ -theorem hold for G . Then the Sylow π -theorem holds for G . This result confirms a conjecture posed by H. Wielandt in 1959.

On the local case in the Aschbacher theorem for linear and unitary groups

We consider the subgroups H in a linear or a unitary group G over a finite field such that Or(H) ≰ Z(G) for some odd prime r. We obtain a refinement of the well-known Aschbacher theorem on subgroups of classical groups for this case.