Search

Let σ = { σ i ∣ i ∈ I } be some partition of the set of all primes and G a finite group. Then G is said to be σ -full if G has a Hall σ i -subgroup for all i ; σ -primary if G is a σ i -group for some i ; σ -soluble if every chief factor of G is σ -primary; σ -nilpotent if G is the direct product of σ -primary groups; G N σ denotes the σ -nilpotent residual of G , that is, the intersection of all normal subgroups N of G with σ -nilpotent quotient G / N . 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 ; σ -subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G such that either A i - 1 ⊴ A i or A i / (A i - 1) A i is σ -primary for all i = 1 , ... , n . Let A σ G be the subgroup of A generated by all σ -permutable subgroups of G contained in A and A σ G be the intersection of all σ -permutable subgroups of G containing A . We prove that if G is a finite σ -soluble group, then the σ -permutability is a transitive relation in G if and only if G N σ avoids the pair (A σ G , A σ G) , that is, G N σ ∩ A σ G = G N σ ∩ A σ G for every σ -subnormal subgroup A of G . [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G , that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S -permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S -permutable subgroups of G containing A. We prove that if G is a soluble group, then S -permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$ , that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G. [ABSTRACT FROM AUTHOR]

Let be a finite group and let be the lattice of all subnormal subgroups of . Let and be subgroups of and let be a sublattice in ; i.e., , for all . Then: is the -closure of in ; i.e., the intersection of all subgroups in which includes and is the -core of in , i.e., the subgroup in generated by all subgroups of belonging to . A subgroup is an - -subgroup in if either or and avoids each composition factor of between and ; i.e., . Using these notions, we give some new characterizations of soluble and supersoluble subgroups and generalize a few available results. [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. A subgroup A of a finite group G is said to be: (i) σ-subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G such that either A i − 1 ⊴ A i or A i / (A i − 1) A i is a σ j -group, j = j (i) , for all i = 1 , ... , n ; (ii) modular in G if the following conditions are held: (1) 〈 X , A ∩ Z 〉 = 〈 X , A 〉 ∩ Z for all X ≤ G , Z ≤ G such that X ≤ Z , and (2) 〈 A , Y ∩ Z 〉 = 〈 A , Y 〉 ∩ Z for all Y ≤ G , Z ≤ G such that A ≤ Z ; (iii) σ-quasinormal in G if A is σ -subnormal and modular in G. We obtain a description of finite groups in which σ -quasinormality (respectively, modularity) is a transitive relation. Some known results are extended. [ABSTRACT FROM AUTHOR]

Let G be a finite group and ℒ(G) the subgroup lattice of G. A subgroup M of G is called: (i) modular in G, if M is a modular element (in the sense of Kurosh) of the lattice ℒ(G); (ii) submodular in G if G has a chain of subgroups M = M0 ≤ M0 ≤⋯ ≤ Mt = G, where Mi is modular in Mi+1 for all i = 0, 1,…,t − 1. If A is a subgroup of G, then we denote by AmG the subgroup of A, generated by all of its subgroups that are modular in G.We say that a subgroup A is N-modular in G (N ≤ G), if for some modular subgroup T of G, containing A, N avoids the pair (T,AmG), i.e. N ∩ T = N ∩ AmG.We prove that if G is a soluble finite group and each of its submodular subgroups is N-modular in G, where N is the nilpotent residual of G, then the lattice ℒ(G) is modular. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite and G always denotes a finite group; L(G) is the lattice of all subgroups of G; σ = {σι | i ∈ I} is some partition of the set of all primes. A subgroup A of G is said to be: (i) o-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤...≤ An = G such that either Ai-1... Ai or Ai/(Ai-1)A; is o-primary for all i = 1,..., n; (ii) o-permutable in G if G is σ-full, that is, G has a Hall σi-subgroup for every σi ∈ σ(G) and A permutes with all such Hall subgroups of G. In this review we discuss some new results on finite groups with given systems σ-subnormal and σ-permutable subgroups. In particular, we discuss groups in which σ-permutability is a transitive relation in the group. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite. Let be some partition of the set of all primes. If n is an integer, the symbol denotes the set ; and. We call any function f of the form a formation σ-function, and we put If for some formation σ-function f we have , then we say that the class is σ-local and f is a σ-local definition of. We suppose that every formation is 0-multiply σ-local; for n > 0, we say that the formation is n-multiply σ-local provided either is the class of all identity groups or where is multiply σ-local for all. In this paper, we describe some properties and examples of n-multiply σ-local formations. In particular, we prove that the Gaschütz product of any two n-multiply σ-local formations is also n-multiply σ-local. We also consider one application of such formations in the theory of finite factorizable groups. [ABSTRACT FROM AUTHOR]

Let σ = { σ i ∣ i ∈ I } \sigma=\{\sigma_{i}\mid i\in I\} be some partition of the set of all primes and 퐺 a finite group. Then 퐺 is said to be 휎-full if 퐺 has a Hall σ i \sigma_{i} -subgroup for all i ∈ I i\in I and 휎-primary if 퐺 is a σ i \sigma_{i} -group for some 푖. In addition, 퐺 is 휎-soluble if every chief factor of 퐺 is 휎-primary and 휎-nilpotent if 퐺 is a direct product of 휎-primary groups. We write G N σ G^{\mathfrak{N}_{\sigma}} for the 휎-nilpotent residual of 퐺, which is the intersection of all normal subgroups 푁 of 퐺 with 휎-nilpotent G / N G/N . A subgroup 퐴 of 퐺 is said to be 휎-permutable in 퐺 provided 퐺 is 휎-full and 퐴 permutes with all Hall σ i \sigma_{i} -subgroups 퐻 of 퐺 (that is, A ⁢ H = H ⁢ A AH=HA ) for all 푖. And 퐴 is 휎-subnormal in 퐺 if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A n = G A=A_{0}\leq A_{1}\leq\cdots\leq A_{n}=G such that either A i − 1 ⁢ ⊴ ⁢ A i A_{i-1}\trianglelefteq A_{i} or A i / ( A i − 1 ) A i A_{i}/(A_{i-1})_{A_{i}} is 휎-primary for all i = 1 , … , n i=1,\ldots,n . We prove that if 퐺 is a 휎-soluble group, then 휎-permutability is a transitive relation in 퐺 if and only if G N σ ∩ A G = G N σ ∩ A G G^{\mathfrak{N}_{\sigma}}\cap A^{G}=G^{\mathfrak{N}_{\sigma}}\cap A_{G} for every 휎-subnormal subgroup 퐴 of 퐺. [ABSTRACT FROM AUTHOR]

It is proved that the lattice of all τ -closed totally saturated formations of finite groups is $ \mathfrak{G} $-separable. [ABSTRACT FROM AUTHOR]

It is proved that the lattice of totally saturated formations of finite groups is distributive. Thus, we give an affirmative answer to the problem proposed by Shemetkov, Skiba and Guo. [ABSTRACT FROM AUTHOR]

We prove that the finite length condition for the lattice of τ-closed totally saturated subformations of a τ-closed totally saturated formation is equivalent to the finite length condition for this lattice, and it is also equivalent for the formation to be soluble and one-generated. [ABSTRACT FROM AUTHOR]

It is proved that the lattice of all τ-closed totally saturated formations of finite groups is algebraic. [ABSTRACT FROM AUTHOR]

We study τ-closed totally saturated formations of finite groups. [ABSTRACT FROM AUTHOR]

It is proved that all proper totally local subformations of a non one-generated totally local formation $\backslash mathfrak\; F$ of finite groups are one-generated iff $\backslash ,\{\backslash mathfrak\; F\}$ coincides with a formation $\{\backslash mathfrak\; S\}\_\{\backslash pi\}$ of all soluble $\backslash pi$-groups, where $|\backslash pi|=2$. [ABSTRACT FROM AUTHOR]