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Let X be a class of groups. Suppose that with each group G ∈ X we associate some system of its subgroups τ (G) . Then τ is said to be a subgroup functor (in the sense of Skiba) on X if the following conditions are hold: (1) G ∈ τ (G) for each group G ∈ X ; (2) for any epimorphism φ : A → B , where A , B ∈ X , and for any groups H ∈ τ (A) and T ∈ τ (B) we have H φ ∈ τ (B) and T φ − 1 ∈ τ (A) . In this paper, we discuss some applications of such subgroup functors in the theory of generalized T-groups. [ABSTRACT FROM AUTHOR]

In this paper, G is a finite group and σ a partition of the set of all primes ℙ, that is, σ = {σi| i ∈ I}, where ℙ = ⋃i∈I σi and σi ∩ σj = ∅ for all i = j; σ (G) = {σi| σi ∩ π(G) = ∅}. We say that G is a σ-tower group if either G = 1 or G has a normal series 1 = G0< G1 < · · · < Gt−1 < Gt= G such that Gk/Gk−1 is a σi-group, σi ∈ σ(G), and G/Gk and Gk−1 are σ′i-groups for all k = 1,... , t. Now we associate with each group G = 1 a directed graph Γ = ΓH σ (G) as follows: Let σ(G) = {σ1 , σ2 ,... , σt}. ΓH σ (G) has t vertices σ1 , σ2 ,... , σt and (σi , σj) is an arc of Γ (directed from σi to σj) if and only if. And we call such a graph the Hawkes σ-graph of G. In this paper, we study various properties of finite Sylow tower and σ-tower groups. In particular, we prove that G is a σ-tower group if and only if the Hawkes σ-graph of G has no circuits, and we describe the structure of a σ-tower group G by appealing to some other properties of the graph ΓHσ (G). [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite and σ is some partition of the set of all primes P (that is, = {i | i 2 I}, where P = [i2Ii and i \ j =; for all i 6= j). A subgroup A of a group G is said to be -subnormal in G if there is a subgroup chain A = A0 A1 · · · An = G such that either Ai-1 E Ai or Ai/(Ai-1)Ai is a j -group for some j = j(i) for all i = 1, . . ., n. In this review, we discuss some known results of the theory of -subnormal sub-groups and also some open questions in this line research. [ABSTRACT FROM AUTHOR]

A finite group G is called a PST-group (respectively, PT-group, T-group) if every subnormal subgroup of G is Sylow permutable (respectively, permutable, normal) in G. Let F be a class of group and G a finite group. Then, a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G ∈ F whenever Σ ⊆ F. We prove that: (i) If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then, Σ is a G-covering subgroup system for the class of all soluble PST-groups; (ii) if Σ is the set of all two-generated subgroups of G, then, Σ is a G-covering subgroup system for the classes of all soluble PST-groups, all soluble PT-groups, and all soluble T-groups. We use these results to prove the following characterizations of soluble PT-groups and T-groups: Suppose that a set of subgroups Σ contains at least one supplement to each maximal subgroup of every Sylow subgroup of G. Then, G is a soluble PT-group (respectively, a soluble T-group) if and only if every subgroup in Σ is a soluble PT-group (respectively, a soluble T-group) and at least one of the nonidentity Sylow subgroups of G is an Iwasawa (respectively, a Dedekind) group. [ABSTRACT FROM AUTHOR]

Throughout this article, all groups are finite and G is a group. Let σ = { σ i | i ∈ I } be some partition of the set of all primes P. Then σ (G) = { σ i | σ i ∩ π (G) ≠ ∅ } ; σ + (G) = { σ i | G has a chief factor H/K such that σ (H / K) = { σ i } }. The group G is said to be: σ-primary if G is a σi-group for some i; σ-soluble if every chief factor of G is σ-primary. The symbol R σ (G) denotes the product of all normal σ-soluble subgroups of G. The chief factor H/K of G is said to be: σ-central (in G) if (H / K) ⋊ (G / C G (H / K)) is σ-primary; a σi-factor if H/K is a σi-group. We say that G is: σ-nilpotent if every chief factor of G is σ-central; generalized { σ i } -nilpotent if every chief σi-factor of G is σ-central. The symbol F { g σ i } (G) denotes the product of all normal generalized { σ i } -nilpotent subgroups of G. We call any function f of the form f : σ ∪ { ∅ } → { formations of groups } , where f (∅) ≠ ∅ , a generalized formation σ-function and we put B L F σ (f) = (G | G / R σ (G) ∈ f (∅) and G / F { g σ i } (G) ∈ f (σ i) for all σ i ∈ σ + (G)). If for some generalized formation σ-function f we have F = B L F σ (f) , then we say that the class F is Baer-σ-local and f is a generalized σ-local definitionof F. In this article, we describe basic properties, examples, and some applications of Baer-σ-local formations. In particular, we prove that the set of all Baer-σ-local formations containing all nilpotent groups forms a subsemigroup of the semigroup of all formations G G (Theorem 1.13) and the set of all Baer-σ-local formations containing all σ-nilpotent groups is a right ideal in G G (Theorem 1.16). [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P and G a finite group. Then G is said to be σ-full if G has a Hall σ i -subgroup for all i ∈ I. A subgroup A of G is said to be σ-permutable in G provided G is σ -full and A permutes with all Hall σ i -subgroups H of G (that is, A H = H A) for all i. We obtain a description of finite σ -full groups G with supersoluble Hall σ i -subgroups for all i ∈ I in which σ -permutability is a transitive relation on G. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite and G always denotes a finite group. Let σ = { σ i | i ∈ I } be a partition of the set of all primes P. The group G is said to be: σ-primary if G is a σ i -group for some i = i (G) ; σ-nilpotent if G = G 1 × ... × G n for some σ -primary groups G 1 , ... , G n ; σ-soluble if every chief factor of G is σ -primary; σ-full if G possesses a Hall σ i -subgroup for all i such that σ i ∩ π (G) ≠ ∅. A subgroup A of G is said to be σ-permutable in G provided G is σ -full and A permutes with every Hall σ i -subgroup H of G , that is, A H = H A for all i ; G is said to be a PσT-group if σ -permutability is a transitive relation in G , that is, if K is a σ -permutable subgroup of H and H is a σ -permutable subgroup of G , then K is a σ -permutable subgroup of G. Let F be a class of group. Then a set Σ of subgroups of G is called a G-covering subgroup system for the class F if G ∈ F whenever Σ ⊆ F. We prove that: If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G, then Σ is a G-covering subgroup system for the classes of all σ-soluble groups, all σ-nilpotent groups, and all σ-soluble PσT-groups. This result gives positive answers to Questions 19.87 and 19.88 in the Kourovka Notebook and, also, allows us to obtain the following characterization of σ -soluble PσT -groups: G is a σ-soluble PσT-group if and only if each maximal subgroup of every Sylow subgroup of G has a supplement T in G such that T is a σ-soluble PσT-group. [ABSTRACT FROM AUTHOR]