The subgroups $A$ and $B$ of a group~$G$ are called {\rm msp}-permutable, if the following statements hold: $AB$~is a subgroup of~$G$; the subgroups $P$ and $Q$ are mutually permutable, where $P$~is an arbitrary Sylow $p$-subgroup of~$A$ and $Q$~is an arbitrary Sylow $q$-subgroup of~$B$, ${p\neq q}$. In the present paper, we investigate groups that factorized by two {\rm msp}-permutable subgroups. In particular, the supersolubility of the product of two supersoluble {\rm msp}-permutable subgroups is proved.
Let $H$ be a subgroup of a group $G$. The permutizer $P_G(H)$ is the subgroup generated by all cyclic subgroups of $G$ which permute with $H$. A subgroup $H$ of a group $G$ is strongly permutable in $G$ if $P_U(H)=U$ for every subgroup $U$ of $G$ such that~$H\le U\le G$. We investigate groups with $\mathbb{P}$-subnormal or strongly permutable Sylow and primary cyclic subgroups. In particular, we prove that groups with all strongly permutable primary cyclic subgroups are supersoluble.
Let $R$ be a subset of a group $G$. We call a subgroup $H$ of $G$ the $R$-conjugate-permutable subgroup of $G$, if $HH^{x}=H^{x}H$ for all $x\in R$. This concept is a generalization of conjugate-permutable subgroups introduced by T. Foguel. Our work focuses on the influence of $R$-conjugate-permutable subgroups on the structure of finite groups in case when $R$ is the Fitting subgroup or its generalizations $F^{*}(G)$ (introduced by H. Bender in 1970) and $\tilde{F}(G)$ (introduced by P. Shmid 1972). We obtain a new criteria for nilpotency and supersolubility of finite groups which generalize …
Let $σ=\{π_i | i\in I$ and $π_i\capπ_j=\emptyset$ for all $i\neq j\}$ be a partition of the set of all primes into mutually disjoint subsets. In this paper we considered subgroups that permutes with given sets of $π_i$-maximal subgroups for all $π_i\in σ$. In particular we showed that such subgroups forms a sublattice of the lattice of all subgroups of a finite group. As corollaries we obtained some well known results about $S$-permutable subgroups.
We prove that if a group $G=AB$ is the mutually permutable product of the supersoluble subgroups $A$ and $B$, then the supersoluble residual of $G$ coincides with the nilpotent residual of the derived subgroup $G'$.
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…
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…
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}…
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 ${\cal H}$.
The additivity with respect to exact sequences is notoriously a fundamental property of the algebraic entropy of group endomorphisms. It was proved for abelian groups by deeply exploiting their structure. On the other hand, a solvable counterexample was recently found, showing that it does not hold in general. Nevertheless, we give a rather short proof of the additivity of the algebraic entropy for locally finite groups that are either quasihamiltonian or FC-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…
According to T. Foguel a subgroup $H$ of a group $G$ is called conjugate-permutable if $ HH^x=H^xH$ for every $x\in G$. Mingyao Xu and Qinhai Zhang studied finite groups with every subgroup conjugate-permutable (ECP-groups) and asked three questions about them. We gave the answers on these questions. In particular, every group of exponent 3 is ECP-group, there exist non-regular ECP-$3$-groups and the class of all finite ECP-groups is neither formation nor variety.
The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=λ_p(G)$ is called the non-$p$-s…
The Hawkes graph $Γ_H(G)$ of $G$ is the directed graph whose vertex set coincides with $π(G)$ and it has the edge $(p, q)$ whenever $q\inπ(G/O_{p',p}(G))$. The Sylow graph $Γ_s(G)$ of $G$ is the directed graph with vertex set $π(G)$ and $(p, q)$ is an edge of $Γ_s(G)$ whenever $q \inπ(N_G(P)/PC_G(P))$ for some Sylow $p$-subgroup $P$ of $G$. The $N$-critical graph $Γ_{Nc}(G)$ of a group $G$ the directed graph whose vertex set coincides with $π(G)$ such that $(p, q)$ is an edge of $Γ_{Nc}(G)$ whenever $G$ contains a Schmidt $(p, q)$-subgroup, i.e. a Schmidt $\{p, q\}$-subgroup with a normal Sylo…
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…
A subgroup $H$ of a group $G$ is called weakly s-supplemented in $G$ if there is a subgroup $T$ such that $G=HT$ and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are s-permutable in $G$. The influence of primary weakly s-supplemented subgroups on the structure of finite groups is investigated. An open question promoted by Skiba is studied and some known results are generalized.
In this paper the concept of $\mathbb{F}$-functorial of a finite group was introduced. These functorials have many properties of the Fitting subgroup of a soluble group and the generalized Fitting subgroup of a finite group. It was shown that the set of all $\mathbb{F}$-functorials is a complete distributive lattice and the cardinality of this lattice is continuum. The sharp bounds on the generalized Fitting height of mutually permutable product of two subgroups were obtained.
A subgroup $H$ of a finite group $G$ is said to be SS-permutable in $G$ if $H$ has a supplement $K$ in $G$ such that $H$ permutes with every Sylow subgroup of $K$. A finite group $G$ is called an SST-group if SS-permutability is a transitive relation on the set of all subgroups of $G$. The structure of SST-groups is investigated in this paper.
Let $G$ be a finite group and $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}σ_{i}$ and $σ_{i}\cap σ_{j}=\emptyset$ for all $i\neq j$. A chief factor $H/K$ of $G$ is said to be $σ$-central in $G$, if the semidirect product $(H/K)\rtimes(G/C_G(H/K))$ is a $σ_i$-group for some $i\in I$. The group $G$ is said to be $σ$-nilpotent if either $G=1$ or every chief factor of $G$ is $σ$-central. In this paper, we study the properties of a finite group $G=AB$, factorized by two $σ$-nilpotent subgroups $A$ and $B$, and also generalize some k…
Let $\Bbb P$ be the set of all primes. A subgroup $H$ of a group $G$ is called {\it $\mathbb P$-subnormal} in $G$, if either $H=G$, or there exists a chain of subgroups $H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb P, \ \forall i.$ A group $G$ is called {\it widely supersoluble}, $\mathrm{w}$-supersoluble for short, if every Sylow subgroup of $G$ is $\mathbb P$-subnormal in $G$. A group $G=AB$ with $\mathbb P$-subnormal $\mathrm{w}$-supersoluble subgroups $A$ and $B$ is studied. The structure of its $\mathrm{w}$-supersoluble residual is obtained. In particular, it coincides with…
Let $\frak{F}$ be a class of finite groups. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{F_{\mathrm s}}$-quasinormal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable in $G$ and $(H\cap T)H_G/H_G$ is contained in the $\frak{F}$-hypercenter $Z_\infty^\mathfrak{F}(G/H_G)$ of $G/H_G$. In this paper, we investigate further the influence of $\mathfrak{F_{\mathrm s}}$-quasinormality of some subgroups on the structure of finite groups. New characterization of some classes of finite groups are obtained.
We consider groups of the form $G=AB$ with two locally cyclic subgroups $A$ and $B$. The structure of these groups is determined in the cases when $A$ and $B$ are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case when $A$ and $B$ are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic groups. It is also shown that the product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group.
The nonsoluble length $λ(G)$ of a finite group $G$ is defined as the number of nonsoluble factors in a shortest normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if a finite group $G=AB$ is factorized by two subgroups of coprime orders, then the nonsoluble length of $G$ is bounded in terms of…
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…
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 $σ$-…
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…
Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $σ(G)=\{σ_{i} : σ_{i}\cap π(G)\ne \emptyset$. 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 $i\in I$ and $\cal H$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i$ such that $σ_{i}\in σ(G)$. We say that $G$ is $σ$-full if $G$ possesses a complete Hall $σ$-set. A complete Hall $σ$-set $\cal H$ of $G$ is said to be a $σ$-basis of $G$ if every two subgroups $A, B \…
A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, $n$-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups $G = AB$ with subgroups $A,\ B$ such that $\langle a, b\rangle$ is soluble for all $a \in A$ a…
The neural connectome of the nematode Caenorhabditis elegans has been completely mapped, yet in spite of being one of the smallest connectomes (302 neurons), the design principles that explain how the connectome structure determines its function remain unknown. Here, we find symmetries in the locomotion neural circuit of C. elegans, each characterized by its own symmetry group which can be factorized into the direct product of normal subgroups. The action of these normal subgroups partitions the connectome into sectors of neurons that match broad functional categories. Furthermore, symmetry pr…
