Permutable Subgroups Research Articles

Weakly totally permutable products and Fitting classes

It is known that if $ G=AB $ is a product of its totally permutable subgroups $ A $ and $ B $‎, ‎then $ Gin mathfrak{F} $ if and only if $ Ain mathfrak{F} $ and $ Bin mathfrak{F} $ when $ mathfrak{F} $ is a Fischer class containing the class $ mathfrak{U} $ of supersoluble groups‎. ‎We show that this holds when $ G=AB $ is a weakly totally permutable product for a particular Fischer class‎, ‎$ mathfrak{F}diamond mathfrak{N} $‎, ‎where $ mathfrak{F} $ is a Fitting class containing the class $ mathfrak{U} $ and $ mathfrak{N} $ a class of nilpotent groups‎. ‎We also extend some results concerning the $ mathfrak{U} $-hypercentre of a totally permutable product to a weakly totally permutable product‎.

On the supersolubility of a finite group factorized into pairwise permutable seminormal subgroups

A subgroup $A$ of a group $G$ is called <i>seminormal</i> in $G$ if there exists a subgroup $B$ such that $G=AB$ and $AX$&nbsp;is a subgroup of $G$ for every subgroup $X$ of $B$. We study groups $G = G_1\ldots G_n$ with pairwise permutable subgroups $G_1,

О перестановочных строго 2-максимальных и строго 3-максимальных подгруппах

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

Finite groups with given nearly SΦ-embedded subgroups

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. Then [Formula: see text] is said to be [Formula: see text]-permutably embedded in [Formula: see text] if a Sylow [Formula: see text]-subgroup of [Formula: see text] is also a Sylow [Formula: see text]-subgroup of some [Formula: see text]-permutable subgroup of [Formula: see text] for every prime dividing the order of [Formula: see text]. We say that [Formula: see text] is nearly[Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] is [Formula: see text]-permutable in [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutably embedded in [Formula: see text]. In this paper, we study the properties of the nearly [Formula: see text]-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.

On finite factorized groups with permutable subgroups of factors

Two subgroups A and B of a group G are called 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, $${p\ne q}$$ . In the present paper, we investigate groups that are factorized by two msp-permutable subgroups. In particular, the supersolubility of the product of two supersoluble msp-permutable subgroups is proved.

Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups

Abstract 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 using the structure theorems for such groups in an essential way. 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 algebraic entropy for locally finite groups that are either quasihamiltonian or FC-groups.

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.

Permutable subgroups and the Maier-Schmid theorem for nilpotent-by-finite groups

We prove that every core-free permutable subgroup of a nilpotent-by-finite group G is contained in the hypercentre of G. We also discuss some properties of permutable subgroups of arbitrary groups with respect to commutators and to the upper central series.

On strongly $\Pi$-permutable subgroups of a finite group

On strongly $\Pi$-permutable subgroups of a finite group

Solutions of fixed period in the nonlinear wave equation on networks

The wave equation on network is defined by \(\partial _{tt}u=\Delta _{\mathfrak {G}}u+g(u)\), where \(u\in {\mathbb {R}}^{n}\) and the graph Laplacian \(\Delta _{\mathfrak {G}}\) is an operator on functions on n vertices. We suppose that \(g:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is an odd continuous function that satisfies \(g(0)=0\), \(Dg(0)=0\) and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations \(\Gamma \), using a \(\Gamma \)-equivariant topological invariant we prove the existence of multiple non-constant p-periodic solutions characterized by their symmetries.

On the Lattice of all $${{\mathcal {H}}}$$-Permutable Subgroups of a Finite Group

Let G be a finite group and $$\sigma =\{\sigma _{i} | i\in I\}$$ is a partition of all primes $$\mathbb {P}$$. A set $${{{\mathcal {H}}}}$$ of subgroups of G is called 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 $$\sigma _{i}\in \sigma (G)$$; a $$\sigma $$-basis of G if $${{{\mathcal {H}}}}$$ is a complete Hall $$\sigma $$-set of G such that each of the subgroups $$A, B\in {{{\mathcal {H}}}}$$ permutable. For any set $${{\mathcal {H}}}$$ of subgroups of G, we write $${{{\mathcal {P}}}}({{\mathcal {H}}})$$ to denote the set of all subgroups A of G such that A permutes with all subgroups $$B\in {{{\mathcal {H}}}}$$, that is, $$AB=BA$$; $${{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)$$ to denote the set of all subgroups A of G with $$A\in {{{\mathcal {P}}}}({{\mathcal {H}}})$$ and soluble $$A^{G}/A_{G}$$. We prove that the set $${{{\mathcal {P}}}} ({{\mathcal {H}}})$$ and the set $${{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)$$ form sublattices of the lattice $${{{\mathcal {L}}}}(G)$$ of all subgroups of G for each $$\sigma $$-basis $${{\mathcal {H}}}$$ of G. We prove also that if all members of $${{\mathcal {H}}}$$ are nilpotent, then the lattice $${{{\mathcal {L}}}}_{{{\mathcal {H}}}{\mathfrak {S}}}(G)$$ is modular if and only if each of the two subgroups $$A, B \in {{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)$$ are permutable.

On σ-conditionally permutable subgroups of finite groups

Let σ be some partition of the set of all primes and a complete Hall σ-set of a finite group G. A subgroup H of G is said to be σ-conditionally permutable in G if for any subgroup there exists an element such that In this article, we investigate the influence of σ-conditionally permutable subgroups on the structure of finite groups.

On two open problems of the theory of permutable subgroups of finite groups

On two open problems of the theory of permutable subgroups of finite groups

The Fitting length of a product of mutually permutable finite groups

Let the finite soluble group $${G = G_{1}G_{2} \cdots G_{r}}$$ be the product of pairwise mutually permutable subgroups $${G_{1}, G_{2}, \ldots, G_{r}}$$ , let h(G) and $${\ell_{p}(G)}$$ be respectively the Fitting length and the p-length of G. The aim of this paper is to prove that $${h(G) \leq {\rm max} \{h(G_{i}) \mid i = 1, 2, \ldots, r\}+1}$$ and $${\ell_{p}(G) \leq {\rm max} \{\ell_{p}(G_{i}) \mid i = 1, 2, \ldots, r\}+1}$$ .

弱NS&lt;sup&gt;*&lt;/sup&gt;-置换子群对有限群超可解的影响

弱NS&lt;sup&gt;*&lt;/sup&gt;-置换子群对有限群超可解的影响

A note on weakly $\mathfrak{Z}$-permutable subgroups of finite groups

A note on weakly $\mathfrak{Z}$-permutable subgroups of finite groups

On $W$-$S$-permutable subgroups of finite groups

A subgroup H of a finite group G is said to be W - S -permutable in G if there is a subgroup K of G such that G=HK and H\cap K is a nearly S -permutable subgroup of G . In this article, we analyse the structure of a finite group G by using the properties of W - S -permutable subgroups and obtain some new characterizations of finite p -nilpotent groups and finite supersolvable groups. Some known results are generalized

On $H_{\sigma}$-permutably embedded subgroups of finite groups

Let G be a finite group. Let \sigma =\{\sigma_{i} | i\in I\} be a partition of the set of all primes \mathbb P and n an integer. We write \sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \} , \sigma (G) =\sigma (|G|) . A set {\mathcal H} of subgroups of G is said to be a complete Hall \sigma -set of G if every member of {\mathcal H}\setminus \{1\} is a Hall \sigma _{i} -subgroup of G for some \sigma _{i} and {\mathcal cal H} contains exact one Hall \sigma _{i} -subgroup of G for every \sigma _{i}\in \sigma (G) . A subgroup A of G is called (i) a \sigma - Hall subgroup of G if \sigma (A) \cap \sigma (|G:A|)=\emptyset ; (ii) {\sigma} -permutable in G if G possesses a complete Hall \sigma -set {\mathcal H} such that AH^x=H^xA for all H\in {\mathcal H} and all x\in G . We say that a subgroup A of G is H_{\sigma} - permutably embedded in G if A is a {\sigma} -Hall subgroup of some {\sigma} -permutable subgroup of G . We study finite groups G having an H_{\sigma} -permutably embedded subgroup of order |A| for each subgroup A of G . Some known results are generalized.

On the structure of a mutually permutable product of finite groups

Let a finite group \({G = AB}\) be the product of the mutually permutable subgroups A and B. We investigate the structure of G given by conditions on conjugacy class sizes of elements in \({A \cup B}\) . Some recent results are extended.

Permutability in uncountable groups

A subgroup X of a group G is called permutable if $$XY=YX$$ for all subgroups Y of G. It is well known that permutable subgroups play an important role in many problems in group theory, and the purpose of this paper is to describe the structure of uncountable groups of regular cardinality $$\aleph $$ in which all large subgroups are permutable.