Existence criterion for Hall subgroups of finite groups

A Criterion for the Existence of a Solvable $π$-Hall Subgroup in a Finite Group

Let $G$ be a finite group and let $π$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $π$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $π$-Hall subgroup if, and only if, $G$ has a $\{p,q\}$-Hall subgroup for any pair $p$, $q\inπ$.

On the heritability of the Hall property $D_π$ by overgroups of $π$-Hall subgroups

Let $π$ be a set of primes. We say that a finite group $G$ is a $D_π$-group if all its maximal $π$-subgroups are conjugate. In this paper, we give an affirmative answer to the problem 17.44(b) from "The Kourovka notebook": we prove that in a $D_π$-group an overgroup of a $π$-Hall subgroup is always a $D_π$-group.

The Nilpotency Criterion for the Derived Subgroup of a Finite Group

It is proved that the derived subgroup of a finite group is nilpotent if and only if $|ab|\ge |a||b|$ for all primary commutators $a$ and $b$ of coprime orders.

On the intersection of solvable Hall subgroups in finite simple exceptional groups of Lie type

Assume that a finite almost simple group with simple socle isomorphic to an exceptional group of Lie type possesses a solvable Hall subgroup. Then there exist four conjugates of the subgroup such that their intersection is trivial.

On the pronormality of Hall subgroups

Fix a set of primes $π$. A finite group is said to satisfy $C_π$ or, in other words, to be a $C_π$-group, if it possesses exactly one class of conjugate $π$-Hall subgroups. The pronormality of $π$-Hall subgroups in $C_π$-groups is proven, or, equivalently, we prove that $C_π$ is inherited by overgroups of $π$-Hall subgroups. Thus an affirmative solution to Problem 17.44(a) from the "Kourovka notebook" is obtained. We also provide an example, showing that Hall subgroups in finite groups are not pronormal in general.

Pronormality of Hall subgroups in finite simple groups

We prove that Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of "Kourovka notebook".

On the number of classes of conjugate Hall subgroups in finite simple groups

In this paper we find the number of conjugate $π$-Hall subgroups in all finite almost simple groups. We also complete the classification of $π$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.

The Hall property $\mathcal{D}_π$ is inherited by overgroups of $π$-Hall subgroups

Let $π$ be a set of primes. We say that a finite group $G$ is a $\mathcal{D}_π$-group if the maximal $π$-subgroups of $G$ are conjugate. In this paper, we give an affirmative answer to Problem 17.44(b) from "Kourovka notebook", namely we prove that in a $\mathcal{D}_π$-group an overgroup of a $π$-Hall subgroup is always a $\mathcal{D}_π$-group.

Finite groups with permutable Hall subgroups

Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i$ such that $σ_{i}\cap π(G)\ne \emptyset$. In this paper, we study the structure of $G$ assuming that some subgroups of $G$ permutes with all members of ${\cal H}$.

Intersection of Solvable Hall subgroups in finite groups

In this paper we show that if S is a simple classical group, a group G is contained in inner-diagonal automorphisms of S and contains S, and H is a solvable Hall subgroup of G, then there exists five conjugates of H, whose intersection is trivial.

Hall's universal group is a subgroup of the abstract commensurator of a free group

P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.

A conjugacy criterion for Hall subgroups in finite groups

A finite group $G$ is said to satisfy $C_π$ for a set of primes $π$, if $G$ possesses exactly one class of conjugate $π$-Hall subgroups. In the paper we obtain a criterion for a finite group $G$ to satisfy $C_π$ in terms of a normal series of the group.

可解线性群的Hall子群的大轨道

Large orbits of Hall subgroups of solvable linear groups

本文证明有限可解群中Hall子群的最大特征标次数平方至少为其某正规商群的π部分，刻画了可解线性群中Hall子群的大轨道性质。

通过特征标次数刻画正规 Hall 子群，II

Characterizing normal Hall subgroups by character degrees, II

本文证明了有限单经典群中真非平凡 Hall 子群的特征标次数性质，从而结合已有约化，给出了任意有限群中 Hall 子群正规性的充要条件：其诱导表示的不可约成分次数均为 $\pi'$ 数。

基于类大小的正规子群为超中心群的判定准则

A criterion for a normal subgroup to be hypercentral based on class sizes

该研究针对有限群正规子群的结构判定问题，利用包含于正规子群的群共轭类大小与重数信息，给出了固定素数下正规子群的Sylow p-子群及整体是否为群超中心子群的判定准则。

On the lattice of the $σ$-permutable subgroups of a finite group

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $σ(G) =\{σ_{i} |σ_{i}\cap π(G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ if $G$ possesses a complete Hall $σ$-set and $A$ permutes with each Hall $σ_{i}$-subgroup $H$ of $G$, that is, $AH=HA$ fo…

Arad–Ward 关于 Hall 子群定理的一个推广

A generalization of the Arad--Ward theorem on Hall subgroups

本文推广 Arad–Ward 定理，证明同时属于两类 Hall 子群群类的有限群也属于其素数集交对应的群类，并推出在所有适当大小素数子集上均含 Hall 子群时存在可解 Hall $π$-子群。

On $H_σ$-permutably embedded and $H_σ$-subnormaly embedded subgroups of finite groups

Let $G$ be a finite group. Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $n$ an integer. We write $σ(n) =\{σ_{i} |σ_{i}\cap π(n)\ne \emptyset \}$, $σ(G) =σ(|G|)$. A set $ {\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member of ${\cal H}\setminus \{1\}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $A$ of $G$ is called: (i) a $σ$-Hall subgroup of $G$ if $σ(|A|) \cap σ(|G:A|)=\emptyset$; (ii) $σ$-permutable in $G$ if $G…

P. Hall 关于极大子群指数定理的推广

Extensions of a theorem of P. Hall on indexes of maximal subgroups

本文推广 P. Hall 关于极大子群指数与有限群可解性的经典定理，证明在极大子群幂零或真非极大子群含于素数（平方）指数子群时，$G$ 仍可解。

确定有限群中的超中心 Hall 子群

Determining hypercentral Hall subgroups in finite groups

本文研究如何利用有限群中π-元素的共轭类长度及其重数推断群的π-结构，并证明该类长度多重集可判定群是否具有超中心Hall π-子群。

A criterion for solvability of a finite group by the sum of subgroup orders

Let $G$ be a finite group and $σ_1(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$. In this note, we prove that if $σ_1(G)<\frac{117}{20}$, then $G$ is solvable. Moreover, we have $σ_1(G)=\frac{117}{20}$ if and only if $G\cong A_5$. This solves an open problem posed in \cite{8}.

On the existence of Carter subgroups

In the paper we obtain the existence criterion of a Carter subgroup in a finite group in terms of its normal series. An example showing that the criterion cannot be reformulated in terms of composition factors is given.

π-元素的共轭类与幂零/阿贝尔 Hall π-子群

Conjugacy classes of $π$-elements and nilpotent/abelian Hall $π$-subgroups

本文研究有限群中 π-元素共轭类的数量下界，并证明当该数量足够大时，群必含有幂零或阿贝尔的 Hall π-子群。

有限群及其Hall子群的长度参数

Length parameters of finite groups and their Hall subgroups

研究有限群及其Hall子群的长度参数，证明若有限群G有Hallπ-子群H，则G的非p-可解长度受H的广义Fitting高度限制。

On abelian subgroups of finitely generated metabelian groups

In this note we introduce the class of $\mathcal H$-groups (or Hall groups) related to the class of $\mathcal B$-groups defined by Ph. Hall in 1950's. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely-generated metabelian groups. Hall groups allow to give a negative answer to the Baumslag's conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelia…

Artin群的子群可分性 II

Subgroup Separability of Artin Groups II

本研究建立了判定Artin群子群可分性（LERF）的新准则，证明其等价于不含特定诱导子图，推广了Metaftsis-Raptis准则，并得出Artin群为ERF的充要条件及子群可分性与凝聚性的关联。

Nilpotent and Abelian Hall Subgroups in Finite Groups

We give a characterization of the finite groups having nilpotent or abelian Hall $π$-subgroups which can easily be verified from the character table.

On the converse of Hall's theorem

In this paper, we mainly investigate the converse of a well-known theorem proved by P. Hall, and present detailed characterizations under the various assumptions of the existence of some families of Hall subgroups. In particular, we prove that if $p\neq 3$ and a finite group $G$ has a Hall $\{p,q\}$-subgroup for every prime $q\neq p$, then $G$ is $p$-soluble.

On $Π$-permutable subgroups of finite groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $Π$ a non-empty subset of the set $σ$. A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a \emph{ complete Hall $Π$-set} of $G$ if every member of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in Π$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in Π$ such that $σ_i\cap π(G)\neq\emptyset$. A subgroup $H$ of $G$ is called \emph{$Π$-quasinormal} or \emph{$Π$-permutable} in $G$ if $G$ possesses a complete Hall $Π$-set ${\cal H}=\{H_{1}, \ldots , H_{t}…

On $σ$-semipermutable subgroups of finite groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $σ(G) =\{σ_{i} |σ_{i}\cap π(G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $H$ of $G$ is said to be: \emph{$σ$-semipermutable in $G$ with respect to ${\cal H}$} if $HH_{i}^{x}=H_{i}^{x}H$ for all $x\in G$ and all $H_i\in {\cal H}$ such th…

An alternative proof of Hill's criterion of freeness for abelian groups

In this note, we provide a different proof of Hill's criterion of freeness for abelian groups. Our proof hinges on the construction of suitable families of subgroups of the links in Hill's theorem and, ultimately, on the construction of such a family of pure subgroups of the group itself.

Finite index subgroups of fully residually free groups

Using graph-theoretic techniques for f.g. subgroups of $F^{\mathbb{Z}[t]}$ we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. Also we obtain an analogue of Greenberg-Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. non-abelian subgroup of a f.g. fully residually free group is of finite index in its commensurator.

Finite Groups Having Nonnormal T.I. Subgroups

In the present paper, the structure of a finite group $G$ having a nonnormal T.I. subgroup $H$ which is also a Hall $π$-subgroup is studied. As a generalization of a result due to Gow, we prove that $H$ is a Frobenius complement whenever $G$ is $π$-separable. This is achieved by obtaining the fact that Hall T.I. subgroups are conjugate in a finite group. We also prove two theorems about normal complements one of which generalizes a classical result of Frobenius.

Hall π-子群与π'度数的特征

Hall $π$-subgroups and characters of $π'$-degree

本文探讨了有限群中Hall π-子群与π'度数特征标之间的关系，并将Navarro和Tiep的结果推广到多个奇素数的情况。

On weakly $sigma$-permutable subgroups of finite groups

Let G be a finite group and σ = {σ_i, i \in I} be a partition of the set of all primes \mathbb{P}. A set \mathcal{H} of subgroups of G with 1 \in \mathcal{H} is said to be a complete Hall σ-set of G if every non-identity member of \mathcal{H} is a Hall σ_i-subgroup of G. A subgroup H of G is said to be σ-permutable if G possesses a complete Hall σ-set \mathcal{H} such that HA^x = A^xH for all A \in \mathcal{H} and all x \in G. We say that a subgroup H of G is weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H \cap T \leq H_σG. where H_σG is the subgro…

A cohomological criterion for $p$-solvability

Let $G$ be a finite group, $p$ a prime and $P$ a Sylow $p$-subgroup of $G$. In this note we give a cohomological criterion for the $p$-solvability of $G$ depending on the cohomology in degree $1$ with coefficients in $\mathbb F_p$ of both the normal subgroups of $G$ and $P$. As a byproduct we bound the minimal number of quotients of order a power of $p$ appearing in any normal series of $G$ by the number of generators of $P$.

On the intersections of solvable Hall subgroups in finite groups

In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $π$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_π(G)$ holds. The minimal counter example is shown to be an almost simple group of Lie type.

A new criterion for finite non-cyclic groups

Let $H$ be a subgroup of a group $G$. We say that $H$ satisfies the power condition with respect to $G$, or $H$ is a power subgroup of $G$, if there exists a non-negative integer $m$ such that $H=G^{m}=<g^{m} | g \in G >$. In this note, the following theorem is proved: Let $G$ be a group and $k$ the number of non-power subgroups of $G$. Then (1) $k=0$ if and only if $G$ is a cyclic group(theorem of F. Sz$\acute{a}$sz) ;(2) $0 < k <\infty$ if and only if $G$ is a finite non-cyclic group; (3) $k=\infty$ if and only if $G$ is a infinte non-cyclic group. Thus we get a new criterion for the finite …

A bound for Hall's criterion for nilpotence in semi-abelian categories

In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and we improve the bound of Gray's main theorem of [3,Theorem 3.4] (see Main Theorem). And this bound is best possible.

特征标、Hall子群与正规补群

Characters, Hall subgroups, and Normal Complements

本文在更一般假设下，为1989年关于有限群Hall子群的不可约复特征标延拓与正规补群等价的经典结果提供了模版本。

On the inheriting of the property $C_π$ by some normal subgroups

In the paper we prove that the Hall property $C_π$ is inherited by normal subgroups which index is a $π'$-number.

某些言子群的幂零性准则

A nilpotency criterion for some verbal subgroups

本文定义了单换位字，并证明了有限群中对应言子群的幂零性等价于互素阶w-值满足阶乘性条件，同时将该结论推广至w-值集合有限的剩余有限群。

The $D_π$-property on products of $π$-decomposable groups

The aim of this paper is to prove the following result: Let $π$ be a set of odd primes. If the group $G=AB$ is the product of two $π$-decomposable subgroups $A=A_π\times A_{π'}$ and $B=B_π\times B_{π'}$, then $G$ has a unique conjugacy class of Hall $π$-subgroups, and any $π$-subgroup is contained in a Hall $π$-subgroup (i.e. $G$ satisfies property $D_π$).

Hall's Theorem for limit groups

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela.

某些带挠的一关系群中的拟凸面子群

Quasi-convex surface subgroups in some one-relator groups with torsion

本文在特定带挠的一关系群中构造了面子群，并利用这一结果推导了自由群中一个词为原始词的一个profinite准则。

Cyclic subgroup commutativity degrees of finite groups

In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented.

Finite groups whose $n$-maximal subgroups are $σ$-subnormal

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$, for some $i\in I$, and $\cal H$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $H$ of $G$ is said to be: \emph{$σ$-permutable} or \emph{$σ$-quasinormal} in $G$ if $G$ possesses a complete Hall $σ$-set set ${\cal H}$ such that $HA^{x}=A^{x}H$ for all $A\in {\cal H}$ and $x\in G$: \emph{$σ$-subnormal} in $G$ if there i…

On finite $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$σ$-soluble} if every chief factor $H/K$ of $G$ is a $σ_{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $i \in I$ such that $σ_{i}\cap π(G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be \emph{$σ$-permutable} or \emph{$σ$-quasin…

A new solvability criterion for finite groups

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there ex…

A Robinson characterization of finite $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $σ$-full if $G$ has a Hall $σ_{i}$-subgroup for all $i$. 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, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $σ$-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 $σ$-permutabl…

A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter group $F \wr \mathbb{F}_2$ over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on a infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup $L \le…

Characterizations of some classes of finite $σ$-soluble $PσT$-groups

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be $σ$-soluble if every chief factor $H/K$ of $G$ is a $σ_{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in σ$ and ${\cal H}$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i \in I$ such that $σ_{i}\cap π(G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be $σ$-permutable in $G$ if $G$ has a complete Hall $σ$-…

Pseudo Sylow numbers

One part of Sylow's famous theorem in group theory states that the number of Sylow p-subgroups of a finite group is always congruent to 1 modulo p. Conversely, Marshall Hall has shown that not every positive integer $n\equiv 1\pmod{p}$ occurs as the number of Sylow p-subgroups of some finite group. While Hall's proof relies on deep knowledge of modular representation theory, we show by elementary means that no finite group has exactly 35 Sylow 17-subgroups.

On solvable subgroups of automorphism groups of right-angled Artin groups

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.