0000000000047049
AUTHOR
S. F. Kamornikov
On a problem of L.A. Shemetkov on superradical formations of finite groups
Abstract Subgroup-closed saturated formations F which are closed under taking products of F -subnormal F -subgroups are studied in the paper. Our results can be regarded as further developments in the hunt for a solution of a problem proposed by L.A. Shemetkov in 1999 in the Kourovka Notebook.
On formations of finite groups with the generalized Wielandt property for residuals II
A formation [Formula: see text] of finite groups has the generalized Wielandt property for residuals, or [Formula: see text] is a GWP-formation, if the [Formula: see text]-residual of a group generated by two [Formula: see text]-subnormal subgroups is the subgroup generated by their [Formula: see text]-residuals. The main result of this paper describes a large family of GWP-formations to further the transparence of this kind of formations, and it can be regarded as a natural step toward the solution of the classification problem.
On formations of finite groups with the generalised Wielandt property for residuals
Abstract A formation F of finite groups has the generalised Wielandt property for residuals, or F is a GWP-formation, if the F -residual of a group generated by two F -subnormal subgroups is the subgroup generated by their F -residuals. We prove that every GWP-formation is saturated. This is one of the crucial steps in the hunt for a solution of the classification problem.
A note on solubly saturated formations of finite groups
The main aim of this note is to give a criterion for a subgroup-closed formation to be solubly saturated, which we hope may provide a useful proving ground for outstanding questions about this family of formations.
On sigma-subnormal subgroups of factorised finite groups
Abstract Let σ = { σ i : i ∈ I } be a partition of the set P of all prime numbers. A subgroup X of a finite group G is called σ-subnormal in G if there is chain of subgroups X = X 0 ⊆ X 1 ⊆ ⋯ ⊆ X n = G with X i − 1 normal in X i or X i / C o r e X i ( X i − 1 ) is a σ i -group for some i ∈ I , 1 ≤ i ≤ n . In the special case that σ is the partition of P into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality. If a finite soluble group G = A B is factorised as the product of the subgroups A and B, and X is a subgroup of G such that X is σ-subnormal in 〈 X , X g 〉 for all g ∈ A ∪ B , we prove that X is σ-subnormal in G. This is an extension…
On two questions from the Kourovka Notebook
Abstract The aim of this paper is to give answers to some questions concerning intersections of system normalisers and prefrattini subgroups of finite soluble groups raised by the third author, Shemetkov and Vasil'ev in the Kourovka Notebook [10] . Our approach depends on results on regular orbits and it can be also used to extend a result of Mann [9] concerning intersections of injectors associated to Fitting classes.
On complements of 𝔉-residuals of finite groups
ABSTRACTA formation 𝔉 of finite groups has the generalized Wielandt property for residuals, or 𝔉 is a GWP-formation, if the 𝔉-residual of a group generated by two 𝔉-subnormal subgroups is the subgroup generated by their 𝔉-residuals. The main aim of the paper is to determine some sufficient conditions for a finite group to split over its 𝔉-residual.