Finite groups with given systems of generalised σ-permutable subgroups

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G) ≠ ∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ such that AH x = H xA for all H ∈ ℋ and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

Subnormality and residuals for saturated formations: A generalization of Schenkman’s theorem

Let 𝐺 be a finite group, and let 𝔉 be a hereditary saturated formation. We denote by Z F ⁢ ( G ) \mathbf{Z}_{\mathfrak{F}}(G) the product of all normal subgroups 𝑁 of 𝐺 such that every chief factor H / K H/K of 𝐺 below 𝑁 is 𝔉-central in 𝐺, that is, ( H / K ) ⋊ ( G / C G ⁢ ( H / K ) ) ∈ F (H/K)\rtimes(G/\mathbf{C}_{G}(H/K))\in\mathfrak{F} . A subgroup A ⩽ G A\leqslant G is said to be 𝔉-subnormal in the sense of Kegel, or 𝐾-𝔉-subnormal in 𝐺, if there is a subgroup chain A = A 0 ⩽ A 1 ⩽ ⋯ ⩽ A n = G A=A_{0}\leqslant A_{1}\leqslant\cdots\leqslant 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 ∈ F A_{i}/(A_{i-1})_{A_{i}}\in\mathfrak{F} for all i = 1 , … , n i=1,\ldots,n . In this paper, we prove the following generalization of Schenkman’s theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let 𝔉 be a hereditary saturated formation containing all nilpotent groups, and let 𝑆 be a 𝐾-𝔉-subnormal subgroup of 𝐺. If Z F ⁢ ( E ) = 1 \mathbf{Z}_{\mathfrak{F}}(E)=1 for every subgroup 𝐸 of 𝐺 such that S ⩽ E S\leqslant E , then C G ⁢ ( D ) ⩽ D \mathbf{C}_{G}(D)\leqslant D , where D = S F D=S^{\mathfrak{F}} is the 𝔉-residual of 𝑆.

On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups

Let σ={σi:i∈I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is σ-primary if all the prime factors of |G| belong to the same member of σ. G is said to be σ-soluble if every chief factor of G is σ-primary, and G is σ-nilpotent if it is a direct product of σ-primary groups. It is known that G has a largest normal σ-nilpotent subgroup which is denoted by Fσ(G). Let n be a non-negative integer. The n-term of the σ-Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=Fσ(G/Fn(G)). If G is σ-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the σ-nilpotent length of G and it is denoted by lσ(G). If F is a subgroup-closed saturated formation, we define the σ-F-lengthnσ(G,F) of G as the σ-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a σ-soluble, then nσ(A,F)=nσ(G,F)−i for some i∈{0,1,2}.

A note on CAP-subgroups in finite groups

A subgroup [Formula: see text] of a finite group [Formula: see text] is called a CAP-subgroup if [Formula: see text] covers or avoids every chief factor of [Formula: see text]. Let [Formula: see text] be a normal subgroup of a finite group [Formula: see text] and [Formula: see text] be a prime power such that [Formula: see text] or [Formula: see text]. In this note, we show that if all subgroups of order [Formula: see text], and all subgroups of order 4 when [Formula: see text] and [Formula: see text] has a nonabelian Sylow [Formula: see text]-subgroup, of [Formula: see text] are CAP-subgroups of [Formula: see text], then every [Formula: see text]-chief factor of [Formula: see text] is either a [Formula: see text]-group or of order [Formula: see text].

On two sublattices of the subgroup lattice of a finite group

Let {\mathfrak{F}} be a non-empty class of groups, let G be a finite group and let {\mathcal{L}(G)} be the lattice of all subgroups of G. A chief {H/K} factor of G is {\mathfrak{F}} -central in G if {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . Let {\mathcal{L}_{c\mathfrak{F}}(G)} be the set of all subgroups A of G such that every chief factor {H/K} of G between {A_{G}} and {A^{G}} is {\mathfrak{F}} -central in G; {\mathcal{L}_{\mathfrak{F}}(G)} denotes the set of all subgroups A of G with {A^{G}/A_{G}\in\mathfrak{F}} . We prove that the set {\mathcal{L}_{c\mathfrak{F}}(G)} and, in the case when {\mathfrak{F}} is a Fitting formation, the set {\mathcal{L}_{\mathfrak{F}}(G)} are sublattices of the lattice {\mathcal{L}(G)} . We also study conditions under which the lattice {\mathcal{L}_{c\mathfrak{N}}(G)} and the lattice of all subnormal subgroup of G are modular.

Finite groups with only 𝔉 {\mathfrak{F}} -normal and 𝔉 {\mathfrak{F}} -abnormal subgroups

Let G be a finite group, and let {\mathfrak{F}} be a class of groups. A chief factor {H/K} of G is said to be {\mathfrak{F}} -central (in G) if the semidirect product {(H/K)\rtimes(G/C_{G}(H/K))\in\mathfrak{F}} . We say that a subgroup A of G is {\mathfrak{F}} -normal in G if every chief factor {H/K} of G between {A_{G}} and {A^{G}} is {\mathfrak{F}} -central in G and {\mathfrak{F}} -abnormal in G if V is not {\mathfrak{F}} -normal in W for every two subgroups {V<W} of G such that {A\leq V} . We give a description of finite groups in which every subgroup is either {\mathfrak{F}} -normal or {\mathfrak{F}} -abnormal.

On semi-σ-nilpotent finite groups

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A chief factor [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-central if the semidirect product [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. [Formula: see text] is called [Formula: see text]-nilpotent if every chief factor of [Formula: see text] is [Formula: see text]-central. We say that [Formula: see text] is semi-[Formula: see text]-nilpotent (respectively, weakly semi-[Formula: see text]-nilpotent) if the normalizer [Formula: see text] of every non-normal (respectively, every non-subnormal) [Formula: see text]-nilpotent subgroup [Formula: see text] of [Formula: see text] is [Formula: see text]-nilpotent. In this paper we determine the structure of finite semi-[Formula: see text]-nilpotent and weakly semi-[Formula: see text]-nilpotent groups.

ON A LATTICE CHARACTERISATION OF FINITE SOLUBLE PST-GROUPS

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.