Complete Hall Research Articles

An algebraic approach to Harder-Narasimhan filtrations

In this article we study chains of torsion classes in an abelian category A. We prove that chains of torsion classes satisfying mild technical conditions induce a Harder-Narasimhan filtration for every non-zero object M in A, generalising a well-known property of stability conditions. We also characterise the slicings of A in terms of chains of torsion classes. We finish the paper by showing that chains of torsion classes induce wall-crossing formulas in the completed Hall algebra of the category.

Finite groups with some σ-primary subgroups weakly m-ℋ-permutable

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P , G a finite group and σ ( G ) = { σ i | σ i ∩ π ( G ) ≠ ∅ } . A set H of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of H is a Hall σi -subgroup of G for some i and H contains exactly one Hall σi -subgroup of G for every σ i ∈ σ ( G ) . Let H be a complete Hall σ-set of G. A subgroup H of G is said to be H -permutable if HA = AH for every member A ∈ H ; m- H -permutable if H = 〈 A , B 〉 for some modular subgroup A and H -permutable subgroup B of G; weakly m- H -permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H for some m- H -permutable subgroup S of G. In this paper, we study the structure of finite groups G by assuming that some σ-primary subgroups are weakly m- H -permutable in G. Some recent results are generalized and unified.

Characterizations of finite groups with σ-semiembedded subgroups

Let [Formula: see text] be a partition of the set of all primes [Formula: see text], [Formula: see text] a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be: [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] has a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]; [Formula: see text]-semipermutable in [Formula: see text] if [Formula: see text] has a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]-group [Formula: see text] with [Formula: see text]. We say that [Formula: see text] is [Formula: see text]-semiembedded in [Formula: see text] if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a [Formula: see text]-permutable subgroup of [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the subgroup of [Formula: see text] which is generated by all [Formula: see text]-semipermutable subgroups of [Formula: see text] contained in [Formula: see text]. In this paper, we study the influence of [Formula: see text]-semiembedded subgroups on the structure of finite groups. Some known results are generalized and unified.

Finite groups with σ-abnormal or 𝔉-subnormal σ-primary subgroups

Let [Formula: see text] be some partition of the set of all primes [Formula: see text], [Formula: see text] a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-primary if it is a finite [Formula: see text]-group for some [Formula: see text]; [Formula: see text]-abnormal in [Formula: see text] if [Formula: see text] is not [Formula: see text]-primary whenever [Formula: see text] and [Formula: see text] is a maximal subgroup of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-subnormal if there is a chain of subgroups [Formula: see text] such that [Formula: see text] for all [Formula: see text]. This is equivalent to [Formula: see text]. In this paper, we study the structure of a finite group [Formula: see text] in which every [Formula: see text]-primary cyclic subgroups are [Formula: see text]-quasinormal or [Formula: see text]-abnormal. We also describe the structure of a finite group [Formula: see text] with self-normalizing or [Formula: see text]-subnormal [Formula: see text]-primary cyclic subgroups for a subgroup-closed saturated lattice formation [Formula: see text] containing all [Formula: see text]-nilpotent groups.

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.

On weakly σ-semipermutable subgroups of finite groups

Let [Formula: see text] be some partition of the set of all primes [Formula: see text], G be a finite group and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall [Formula: see text]-set of G if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of G for some i and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of G for every [Formula: see text]. Let [Formula: see text] be a complete Hall [Formula: see text]-set of G. A subgroup A of G is said to be [Formula: see text]-semipermutable with respect to [Formula: see text] if [Formula: see text] for all [Formula: see text] and all [Formula: see text] such that [Formula: see text]; [Formula: see text]-semipermutable in G if A is [Formula: see text]-semipermutable in G with respect to some complete Hall [Formula: see text]-set of G. We say that a subgroup H of G is weakly [Formula: see text]-semipermutable in G if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that HT is [Formula: see text]-permutable in G and [Formula: see text], where [Formula: see text] is the subgroup of H generated by all those subgroups of H which are [Formula: see text]-semipermutable in G. In this paper, we study the structure of G under the condition that some subgroups of G are weakly [Formula: see text]-semipermutable in G.

Finite groups with σ-abnormal or σ-subnormal σ-primary subgroups

Let G be a finite group and be a partition of the set of all primes that is, and for all A set of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of is a Hall σi -subgroup of G for some i and contains exactly one Hall σi -subgroup of G for every 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-subnormal in G if there is a subgroup chain such that either is normal in Ai or is σ-primary for all σ-abnormal in G if is not σ-primary whenever and L is a maximal subgroup of K. In this article, we study the structure of a finite group in which σ-primary cyclic subgroups are σ-abnormal or σ-subnormal in G. We also describe the structure of a finite group G which has a complete Hall σ-set such that for any G has a subgroup A of order and A is σ-abnormal or σ-subnormal in G.

On 𝜎-Subnormal Subgroups of Finite 3'-Groups

For any partition 𝜎 of the set ℙ of all primes, it is proved that if every complete Hall set of type 𝜎 for a finite 3'-group G is reducible into a certain subgroup H of G, then H is 𝜎-subnormal in G.

Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups

Let $\sigma=\{{\sigma_i|i\in I}\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. A set $\mathcal{H} $ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $i$ such that $\sigma_i\cap \pi(G)\neq \emptyset$. Let $\tau_{\mathcal{H}}(A)=\{ \sigma_{i}\in \sigma(G)\backslash \sigma(A) \ |\ \sigma(A) \cap \sigma(H^{G})\neq\emptyset$ for a Hall $\sigma_{i}$-subgroup $H\in \mathcal{H}\}$. A subgroup $A$ of $G$ is said to be $\tau_{\sigma}$-permutable or $\tau_{\sigma}$-quasinormal in $G$ with respect to $\mathcal{H}$ if $AH^{x}=H^{x}A$ for all $x\in G$ and $H\in \mathcal{H}$ such that $\sigma(H)\subseteq \tau_{\mathcal{H}}(A)$, and $\tau_{\sigma}$-permutable or $\tau_{\sigma}$-quasinormal in $G$ if $A$ is $\tau_{\sigma}$-permutable in $G$ with respect to some complete Hall $\sigma$-set of $G$. We say that a subgroup $A$ of $G$ is weakly $\tau_{\sigma}$-quasinormal in $G$ if $G$ has a $\sigma$-subnormal subgroup $T$ such that $AT=G$ and $A\cap T\leq A_{\tau_{\sigma}G}$, where $A_{\tau_{\sigma}G}$ is the subgroup generated by all those subgroups of $A$ which are $\tau_{\sigma}$-quasinormal in $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are weakly $\tau_{\sigma}$-quasinormal in $G$.

Finite groups with some σ-primary subgroups sσ-quasinormal

Let G be a finite group, a partition of the set of all primes and A set of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of is a Hall σi-subgroup of G for some and contains exactly one Hall σi-subgroup of G for every G is said to be σ-full if G possesses a complete Hall σ-set. We say a subgroup H of G is sσ-quasinormal (supplement-σ-quasinormal) in G if there exists a σ-full subgroup T of G such that G = HT and H permutes with every Hall σi-subgroup of T for all In this article, we obtain some results about the sσ-quasinormal subgroups and use them to determine the structure of finite groups. In particular, some new criteria of p-nilpotency, solubility, supersolubility of a group are obtained.

On an open problem of σ-soluble groups

Let be some partition of the set of all primes, that is, and for all Let G be a finite group. A set of subgroups of G is said to be a complete Hall σ-set of G if every nonidentity member of is a Hall σi-subgroup of G and contains exactly one Hall σi-subgroup of G for every G is said to be a σ-group if it possesses a complete Hall σ-set. A σ-group G is said to be σ-dispersive provided G has a normal series and a complete Hall σ-set such that for all In this article, we give a characterization of σ-dispersive group, which gives a positive answer to an open problem of Skiba.

О σ -субнормальных подгруппах конечных 3'-групп

УДК 512.542 Для любого разбиения σ множества всех простых чисел доказывается, что каждое полное холлово множество типа σ конечной 3'-группы G редуцируется в некоторую подгруппу H группы G , то подгруппа H является σ -субнормальной в G .

On σ-Subnormal Subgroups of Finite Groups

Let p be a prime and let σ = {{p}, {p}′} be a partition of the set ℙ of all primes. We prove the following conjecture by Skiba: If each complete Hall set of type σ in a finite group G is reducible to some subgroup H of G then H is σ-subnormal in G.

Finite groups with weakly m-σ-permutable subgroups

Let [Formula: see text] be a finite group, [Formula: see text] be a partition of the set of all primes [Formula: see text] and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] possesses a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]. Let [Formula: see text] be a subgroup of [Formula: see text]. [Formula: see text] is: [Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for some modular subgroup [Formula: see text] and [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text]; weakly[Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if there are an [Formula: see text]-[Formula: see text]-permutable subgroup [Formula: see text] and a [Formula: see text]-subnormal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we investigate the influence of weakly [Formula: see text]-[Formula: see text]-permutable subgroups on the structure of finite groups.

A new characterization of finite σ-soluble PσT-groups

Let \(\sigma =\{\sigma_{i} \mid i\in I\}\) be a partition of the set of all primes \(\mathbb{P}\) and \(G\) a finite group. \(G\) is said to be \emph{\(\sigma\)-soluble} if every chief factor \(H/K\) of \(G\) is a \(\sigma_{i}\)-group for some \(i=i(H/K)\). A set \({\mathcal H}\) of subgroups of \(G\) is said to be a complete Hall \(\sigma \)-set of \(G\) if every member \(\ne 1\) of \({\mathcal H}\) is a Hall \(\sigma_{i}\)-subgroup of \(G\) for some \(\sigma_{i}\in \sigma \) and \({\mathcal H}\) contains exactly one Hall \(\sigma_{i}\)-subgroup of \(G\) for every \(i\) such that \(\sigma_{i}\cap \pi (G)\ne \varnothing\). A subgroup \(A\) of \(G\) is said to be \({\sigma}\)-quasinormal or \({\sigma}\)-permutable in \(G\) if \(G\) has a complete Hall \(\sigma\)-set \(\mathcal H\) such that \(AH^{x}=H^{x}A\) for all \(x\in G\) and all \(H\in \mathcal H\). We obtain a new characterization of finite \(\sigma\)-soluble groups \(G\) in which \(\sigma\)-permutability is a transitive relation in \(G\).

The Influence of Weakly $${\sigma }$$-Permutably Embedded Subgroups on the Structure of Finite Groups

Let $$\sigma =\{{\sigma _i|i\in I}\}$$ be some partition of the set of all primes $${\mathbb {P}}$$ , G a finite group and $$\sigma (G)=\{{\sigma _i|\sigma _i \cap \pi (G) \ne \emptyset }\}$$ . A set $${\mathcal {H}} $$ of subgroups of G is said to be a complete Hall $$\sigma $$ -set of G if every non-identity member of $${\mathcal {H}}$$ is a Hall $$\sigma _i$$ -subgroup of G and $${\mathcal {H}}$$ contains exactly one Hall $$\sigma _i$$ -subgroup of G for every $$\sigma _i\in \sigma (G)$$ . G is said to be $$\sigma $$ -full if G possesses a complete Hall $$\sigma $$ -set. A subgroup H of G is said to be $$\sigma $$ -permutable in G provided there is a complete Hall $$\sigma $$ -set $${\mathcal {H}}$$ of G such that $$HA^x=A^xH$$ for all $$A\in {\mathcal {H}}$$ and all $$x\in G$$ ; $$\sigma $$ -permutably embedded in G if H is $$\sigma $$ -full and for every $$\sigma _i \in \sigma (H)$$ , every Hall $$\sigma _i$$ -subgroup of H is also a Hall $$\sigma _i$$ -subgroup of some $$\sigma $$ -permutable subgroup of G. We call that a subgroup H of G is weakly $${\sigma }$$ -permutably embedded in G if there exists a $$\sigma $$ -subnormal subgroup T of G such that $$G=HT$$ and $$H\cap T\le H_{\sigma eG}$$ , where $$H_{\sigma eG}$$ is the subgroup of H generated by all those subgroups of H which are $$\sigma $$ -permutably embedded in G. In this paper, we study the structure of G under the condition that some given subgroups of G are weakly $${\sigma }$$ -permutably embedded in G. Some known results are generalized.

On weakly 𝓗-permutable subgroups of finite groups

Abstract Let σ = {σi ∣i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi ∣σi ∩ π(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H 0 ≤ H 1 ≤ … ≤ Ht = G such that either H i−1 is normal in Hi or Hi /(H i−1) Hi is σ-primary for all i = 1, …, t. A set 𝓗 of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of 𝓗 is a Hall σi -subgroup of G for some i and 𝓗 contains exactly one Hall σi -subgroup of G for every σi ∈ σ(G). Let 𝓗 be a complete Hall σ-set of G. A subgroup H of G is said to be 𝓗-permutable if HA = AH for all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H 𝓗, where H 𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable. By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.

On Strongly Π-Permutable Subgroups of a Finite Group

Let σ = {σi | i ∈ I} be some partition of the set of all primes ℙ,let ∅ ≠ Π ⊆ σ, and let G be a finite group. A set ℋ of subgroups of G is said to be a complete Hall Π-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some σi ∈ Π and ℋ has exactly one Hall σi-subgroup of G for every σi ∈ Π such that σi ∩ π(G) ≠ ∅. A subgroup A of G is called (i) Π-permutable in G if G has a complete Hall Π-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G; (ii) 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 a σk-group for some k for all i = 1,…,t; and (iii) strongly Π-permutable if A is Π-permutable and σ-subnormal in G. We study the strongly Π-permutable subgroups of G. In particular, we give characterizations of these subgroups and prove that the set of all strongly Π-permutable subgroups of G forms a sublattice of the lattice of all subgroups of G.

Finite Generalized Soluble Groups

Let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and G a finite group. Suppose σ(G) = {σi | σi ∩ π(G) ≠ = ∅}. A set ℋ of subgroups of G is called a complete Hall σ-set of G if every nontrivial member of ℋ is a σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ∈ σ(G). A group G is σ-full if G possesses a complete Hall σ-set. A complete Hall σ-set ℋ of G is called a σ-basis of G if every two subgroups A, B ∈ ℋ are permutable, i.e., AB = BA. In this paper, we study properties of finite groups having a σ-basis. It is proved that if G has a σ-basis, then G is generalized σ-soluble, i.e, |σ(H/K)| ≤ 2 for every chief factor H/K of G. Moreover, it is shown that every complete Hall σ-set of a σ-full group G forms a σ-basis of G iff G is generalized σ-soluble, and for the automorphism group G/CG(H/K) induced by G on any its chief factor H/K, we have |σ(G/CG(H/K))| ≤ 2 and also σ(H/K) ⊆ σ(G/CG(H/K)) in the case |σ(G/CG(H/K))| = 2.

On σ-Embedded and σ-n-Embedded Subgroups of Finite Groups

Let G be a finite group, and let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and σ(G) = {σi | σi ∩ π(G) ≠ ∅}. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ℋ is a Hall σi-subgroup of G and ℋ has exactly one Hall σi-subgroup of G for every σi ∈ σ (G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ℋ such that HAx = AxH for all A ∈ ℋ and x ∈ G. A subgroup H of G is said to be σ-n-embedded in G if there exists a normal subgroup T of G such that HT = HG and H ∩ T ≤ HσG, where HσG is the subgroup of H generated by all those subgroups of H that are σ-permutable in G. A subgroup H of G is said to be σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = HσG and H ∩ T ≤ HσG, where HσG is the intersection of all σ-permutable subgroups of G containing H. We study the structure of finite groups under the condition that some given subgroups of G are σ-embedded and σ-n-embedded. In particular, we give the conditions for a normal subgroup of G to be hypercyclically embedded.