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.     

On the $${{\varvec{}}}$$-length of the mutually ermutable roduct of two $${{\varvec{}}}$$-soluble grous

Let a finite group \(G=AB\) be the mutually permutable product of two p-soluble subgroups A and B for some prime p. We give a bound of the p-length of G from the p-lengths of A and B.     

The influence of <Emphasis Type="Italic">s</Emphasis>-semipermutable subgroups on the structure of finite groups

A subgroup H of a group G is called s-semipermutable in G if H is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. In this paper, we use s-semipermutable subgroups to determine the structure of finite groups. Some of the previous results are generalized.     

On the permutability of sylow subgroups with Schmidt subgroups

A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Sufficient conditions are established for the p-solvability of a finite group in which a Sylow p-subgroup is permutable with some Schmidt subgroups. Sufficient conditions for the solvability of a finite group in which some Schmidt subgroups are permutable are also obtained.     

On <Emphasis Type="Italic">c</Emphasis><Superscript>#</Superscript>-normal subgroups infinite groups

A subgroup H of a finite group G is called a c^#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is a CAP-subgroup of G: In this paper, we investigate the influence of fewer c^#-normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.     

Different prime <Emphasis Type="Italic">N</Emphasis>-ideals and IFP <Emphasis Type="Italic">N</Emphasis>-Ideals

We provide some characterizations of completely prime (completely semiprime) and 3-prime (3-semiprime) N-groups. The relationship between a 3-prime (completely prime) N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N is investigated. Moreover, the notion of IFP N-ideal is defined. We prove that the concept of IFP N-ideal occurs naturally where N is a left permutable (left self distributive, subcommutative) near-ring and Γ a monogenic N-group. Also, we obtain some relationships between an IFP N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N.     

Finite groups with <Emphasis Type="Italic">C</Emphasis>-quasinormal subgroups

Consider some finite group G and a finite subgroup H of G. Say that H is c-quasinormal in G if G has a quasinormal subgroup T such that HT = G and T ∩ H is quasinormal in G. Given a noncyclic Sylow subgroup P of G, we fix some subgroup D such that 1 < |D| < | P| and study the structure of G under the assumption that all subgroups H of P of the same order as D, having no supersolvable supplement in G, are c-quasinormal in G.     

<Emphasis Type="Italic">c</Emphasis>-Semipermutable subgroups of finite groups

A subgroup is called c-semipermutable in G if A has a minimal supplement T in G such that for every subgroup T ₁ of T there is an element x ∈ T satisfying AT ₁ ^x = T ₁ ^x A. We obtain a few results about the c-semipermutable subgroups and use them to determine the structures of some finite groups.     

Products of F*(<Emphasis Type="Italic">G</Emphasis>)-subnormal subgroups of finite groups

A subgroup H of a finite group G is called F*(G)-subnormal if H is subnormal in HF*(G). We show that if a group Gis a product of two F*(G)-subnormal quasinilpotent subgroups, then G is quasinilpotent. We also study groups G = AB, where A is a nilpotent F*(G)-subnormal subgroup and B is a F*(G)-subnormal supersoluble subgroup. Particularly, we show that such groups G are soluble.     

The structure of finite groups with weakly <Emphasis Type="Italic">c</Emphasis>-normal subgroups

For a subgroup of a finite group we introduce a new property called weakly c-normal. Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly c-normal in G if there exists a subnormal subgroup K of G such that \(G=HK\) and \(H\cap K\) is s-quasinormally embedded in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is weakly c-normal in G. Some recent results are generalized and unified.     

On <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript>-supplemented subgroups of finite groups

Suppose that F is a formation of finite groups. We introduce the concept of F _h -supplemented subgroups and investigate the structure of finite groups on assuming that some maximal subgroups of Sylow subgroups, maximal subgroups, minimal subgroups, and 2-maximal subgroup are F _h -supplemented, respectively. Some available results are generalized.     

Finite <Emphasis Type="Italic">p</Emphasis>-groups whose non-normal subgroups have few orders

Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^M(G) and p^m(G) to denote the maximum and minimum of the orders of the non-normal subgroups of G; respectively. In this paper, we classify groups G such that M(G) < 2m(G)?1: As a by-product, we also classify p-groups whose orders of non-normal subgroups are p^k and p^k+1.     

Finite groups whose all proper subgroups are <Emphasis Type="Italic">C</Emphasis>-groups

A group G is said to be a C-group if for every divisor d of the order of G, there exists a subgroup H of G of order d such that H is normal or abnormal in G. We give a complete classification of those groups which are not C-groups but all of whose proper subgroups are C-groups.     

On <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings of nearly bipartite graphs

Let G be a simple graph, let d(v) denote the degree of a vertex v and let g be a nonnegative integer function on V (G) with 0 ≤ g(v) ≤ d(v) for each vertex v ∈ V (G). A g _c -coloring of G is an edge coloring such that for each vertex v ∈ V (G) and each color c, there are at least g(v) edges colored c incident with v. The g _c -chromatic index of G, denoted by χ′g _c (G), is the maximum number of colors such that a gc-coloring of G exists. Any simple graph G has the g _c -chromatic index equal to δ _g (G) or δ _g (G) ? 1, where \({\delta _g}\left( G \right) = \mathop {\min }\limits_{v \in V\left( G \right)} \left\lfloor {d\left( v \right)/g\left( v \right)} \right\rfloor \). A graph G is nearly bipartite, if G is not bipartite, but there is a vertex u ∈ V (G) such that G ? u is a bipartite graph. We give some new sufficient conditions for a nearly bipartite graph G to have χ′g _c (G) = δ _g (G). Our results generalize some previous results due to Wang et al. in 2006 and Li and Liu in 2011.     

The complete reducibility of some <Emphasis Type="Italic">GF</Emphasis>(2)<Emphasis Type="Italic">A</Emphasis><Subscript>7</Subscript>-modules

It is proved that, if G is a finite group with a nontrivial normal 2-subgroup Q such that G/Q ～= A ₇ and an element of order 5 from G acts freely on Q, then the extension G over Q is splittable, Q is an elementary abelian group, and Q is the direct product of minimal normal subgroups of G each of which is isomorphic, as a G/Q-module, to one of the two 4-dimensional irreducible GF(2)A ₇-modules that are conjugate with respect to an outer automorphism of the group A ₇.     

A REALIZATION OF CERTAIN MODULES FOR THE <Emphasis Type="Italic">N</Emphasis> = 4 SUPERCONFORMAL ALGEBRA AND THE AFFINE LIE ALGEBRA <Emphasis Type="Italic">A</Emphasis><Stack><Subscript>2</Subscript><Superscript>(1)</Superscript></Stack>

We shall first present an explicit realization of the simple N = 4 superconformal vertex algebra L _c ^N?=?4 with central charge c = ?9. This vertex superalgebra is realized inside of the bcβγ system and contains a subalgebra isomorphic to the simple affine vertex algebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). Then we construct a functor from the category of L _c ^N?=?4 -modules with c = ?9 to the category of modules for the admissible affine vertex algebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). By using this construction we construct a family of weight and logarithmic modules for L _c ^N?=?4 and L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \). We also show that a coset subalgebra L _A1 \( \left(-\frac{3}{2}{\varLambda}_0\right) \) is a logarithmic extension of the W(2; 3)-algebra with c = ?10. We discuss some generalizations of our construction based on the extension of affine vertex algebra L _A1 (kΛ ₀) such that k + 2 = 1/p and p is a positive integer.     

On <Emphasis Type="Italic">MP</Emphasis>-closed saturated formations of finite groups

A class of groups F is called MP-closed, if it contains every group G = AB such that the F-subgroup A permutes with every subgroup of B and the F-subgroup B permutes with every subgroup of A. We prove that the formation F that contains the class of all supersoluble groups is MP-closed if and only if the formation F(p) is MP-closed for any prime number p, where F is a maximal inner local screen of F. In particular, we prove that the formation of all groups with supersoluble Schmidt subgroups is MP-closed.     

On <Emphasis Type="Italic">G</Emphasis>-covering subgroup systems of finite groups

Let \(\mathcal{F}\) be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for  \(\mathcal{F}\) if \(G\in \mathcal{F}\) whenever \(\Sigma \subseteq \mathcal{F}\). Let p be any prime dividing |G| and P a Sylow p-subgroup of G. Then we write Σ _p to denote the set of subgroups of G which contains at least one supplement to G of each maximal subgroup of P. We prove that the sets Σ _p and Σ _p ∪Σ _q , where q≠p, are G-covering subgroup systems for many classes of finite groups.     

Intersections of two nilpotent subgroups in finite groups with socle <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

Given a finite group G with socle isomorphic to L ₂(q), q ≥ 4, we describe, up to conjugacy, all pairs of nilpotent subgroups A and B of G such that A ∩ B ^g ≠ 1 for all g ∈ G.     

A cohomological criterion for <Emphasis Type="Italic">p</Emphasis>-solvability

Let G be a finite group, let p be a prime, and let P be 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 minimum possible number of factors of p-power order appearing in any normal series of G, in which each factor is either a p-group, a p’-group, or a non-p-solvable characteristically simple group, by the number of generators of P.