Lev Kazarin

On Sylow normalizers of finite groups

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

On the product of two π-decomposable soluble groups

Let the group G = AB be a product of two π-decomposable sub-groups A = Oπ(A) × Oπ′ (A) and B = Oπ(B) × Oπ′ (B) where π is a set of primes. The authors conjecture that Oπ(A)Oπ(B) = Oπ(B)Oπ(A) if π is a set of odd primes. In this paper it is proved that the conjecture is true if A and B are soluble. A similar result with certain additional restrictions holds in the case 2 ∈ π. Moreover, it is shown that the conjecture holds if Oπ ′(A) and Oπ′(B) have coprime orders.

On finite products of soluble groups

Let the finite groupG =AB be the product of two soluble subgroupsA andB, and letπ be a set of primes. We investigate under which conditions for the maximal normalπ-subgroups ofA, B andG the following holds:Oπ(G) ∩Oπ(G) ⊆Oπ(G).

Large subgroups of a finite group of even order

It is shown that if G is a group of even order with trivial center such that |G| > 2|CG(t)| for some involution t ∈ G, then there exists a proper subgroup H of G such that |G| < |H|2. If |G| > |CG(t)| and k(G) is the class number of G, then |G| ≤ k(G)3.

On the Soluble Graph of a Finite Simple Group

The maximal independent sets of the soluble graph of a finite simple group G are studied and their independence number is determined. In particular, it is shown that this graph in many cases has an independent set with three vertices.