Narrow Results

In this paper, the norm $L_i(G)$ of the lower central series in a finite group $G$ is introduced, which unifies the norm of derived subgroups and nilpotent residuals. Some propositions of $L_i(G)$ are obtained, and some related subgroups as well as their equivalent propositions can also be found.

$G$ always represents a finite group. ${ℱ}$ is a formation meeting the condition ${𝒜}\subseteq {ℱ}\subseteq {𝒩},$ where ${𝒩}$ and ${𝒜}$, respectively, represent the class of nilpotent groups and abelian groups. ${𝒦}$ is a subgroup-closed formation. We call ${\widetilde{{𝒩}}}^{{ℱ}{𝒦}}(G)$ the ${ℱ}{𝒦}$-norm of $G$ if ${\widetilde{{𝒩}}}^{{ℱ}{𝒦}}(G)$ is the intersection of the normalizers of $(H^{{ℱ}}{O}^{{𝒦}}(G))$ for all the subgroups $H$ of $G$ satisfying $H^{{ℱ}}\in {𝒩}.$ We mainly demonstrate that ${\widetilde{{𝒩}}}^{{ℱ}{𝒦}}(G)={{𝒩}}^{{ℱ}{𝒦}}(G).$ We also show that $G={{𝒩}}^{{ℱ}{𝒦}}(G)$ if and only if $H={{𝒩}}^{{ℱ}{𝒦}}(H)$ for every subgroup $H$ of $G$ generated by three elements. Furthermore, we introduce the ${{𝒩}}^{{ℱ}{𝒦}}$-groups and $IO$-${{𝒩}}^{{ℱ}{𝒦}}$-groups.

Let $𝔛$ be a class of finite groups. A group $G$ is called a minimal non-$𝔛$-group or simply an $𝔛$-critical group, if $G$ is not in $𝔛$ but all proper subgroups of $G$ are in $𝔛$. An $𝔑$-critical group, for the class $𝔛=𝔑$ all finite nilpotent groups, is called a Schmidt group. We say that a formation $𝔉$ has the Shemetkov property if every $𝔉$-critical group is either a Schmidt group or a cyclic group of prime order. In this paper, properties of the lattice of all soluble subgroup-closed local formations with Shemetkov property are investigated.

Let $G$ be a finite group. We denote by ${\nu }_1(G)$ the commuting probability that two randomly chosen elements of $G$ are commuting. In this paper, we present a complete isoclinic classification of finite non-nilpotent group $G$ with ${\nu }_1(G)>\frac{5}{16}$. It is an improvement on a result of Lescot et al. in 2014.

Let $G$ be a finite group and ${ℒ}(G)$ the subgroup lattice of $G$. A subgroup $M$ of $G$ is called: (i) modular in $G$, if $M$ is a modular element (in the sense of Kurosh) of the lattice ${ℒ}(G)$; (ii) submodular in $G$ if $G$ has a chain of subgroups $M=M_0\le M_0\le \cdots \le M_t=G$, where $M_i$ is modular in $M_{i+1}$ for all $i=0,1,\dots ,t-1$. If $A$ is a subgroup of $G$, then we denote by $A_{mG}$ the subgroup of $A$, generated by all of its subgroups that are modular in $G$.

We say that a subgroup $A$ is $N$-modular in $G$ ($N\le G$), if for some modular subgroup $T$ of $G$, containing $A$, $N$avoids the pair$(T,A_{mG})$, i.e. $N\cap T=N\cap A_{mG}$.

We prove that if $G$ is a soluble finite group and each of its submodular subgroups is $N$-modular in $G$, where $N$ is the nilpotent residual of $G$, then the lattice ${ℒ}(G)$ is modular.

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is $\rho$-semipermutable in $G$ if $H$ permutes with all Sylow subgroups $G_p$ of $G$ such that $(|H|,p)=1$ and $p\left|\right|H^G|$. This paper aims to study the structure of finite groups $G$ in the context of the assumption that all maximal subgroups of a Sylow $p$-subgroup of $G$ are $\rho$-semipermutable and some new results are obtained.

A subgroup $A$ of a finite group $G$ is called a local covering subgroup of $G$ if $A^G=AB$ for all maximal $G$-invariant subgroups $B$ of $A^G$. In this note, we characterize the finite groups $G$ in which all subgroups of order $p^d$ enjoy the local covering property, where $p^d$ is a prime power with $|G{|}_p\ge p^d\ge p$.

In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite-dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite-dimensional positive Archimedean representations of a finite group and establish that, up to order equivalence, these are order direct sums, with unique multiplicities, of the order indecomposable positive representations naturally associated with transitive G-spaces. Character theory is shown to break down for positive representations. Induction and systems of imprimitivity are introduced in an ordered context, where the multiplicity formulation of Frobenius reciprocity turns out not to hold.

A theorem of Friedl and Vidussi says that any 3-manifold $N$ and any non-fibered class in $H^1(N;\mathbb{Z})$ there exists a representation such that the corresponding twisted Alexander polynomial is zero. However, it seems that no concrete example of such a representation is known so far. In this paper, we provide several explicit examples of non-fibered knots and their representations with zero twisted Alexander polynomial.

We study an effective gauge theory whose gauge group is a semidirect product G = G_c ⋊ Γ with G_c and Γ being a connected Lie group and a finite group, respectively. The semidirect product is defined through a projective homomorphism (i.e. homomorphism up to the center of G_c) from Γ into G_c. To be specific, we take SU(3)_L as G_c and ℤ₃ × ℤ₃ as Γ. We notice that the irreducible representations of the gauge group G necessarily contain three G_c-multiplets in spite of the Abelian nature of Γ = ℤ₃ × ℤ₃. This triplication phenomenon is due to the semidirect product structure of G. We suggest that the appearance of three families is attributable to this triplication. We give a toy model on the lepton mixing and show that under a particular vacuum alignment the tri-bimaximal mixing matrix is reproduced.

The solubility graph associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct elements $x$ and $y$ if and only if $〈x,y〉$ is a soluble subgroup of $G$. We examine some properties of solubility graphs.

In this paper, we introduce the concept of a pseudo-conjugation action of a group G on itself. This action behaves like conjugation. The purpose of this paper is twofold. First, it provides several useful properties of pseudo-conjugation actions that should simplify other calculations. Second, it extends specific concepts of group theory, such as AC-groups and the notion of isoclinism between two groups, among others. A connection between a pseudo-conjugation action and the conjugation action is also examined.

The structure and embedding of subgroups permuting with the system normalizers of a finite soluble group are studied in the paper. It is also proved that the class of all finite soluble groups in which every subnormal subgroup permutes with the Sylow subgroups is properly contained in the class of all soluble groups whose subnormal subgroups permute with the system normalizers while this latter is properly contained in the class of all supersoluble groups.

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut(J₂) and Aut(McL), and the automorphism group of PSL(2, 2ⁿ) where n=2^s are also uniquely determined by their order components. Also we discuss about the characterizability of Aut(PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

Let B be the Burnside ring considered as a globally-defined Mackey functor, and k be a field of positive characteristic q. We prove that, when k ⊗_ℤ B is restricted to nilpotent groups of order prime to q, the only simple Mackey functors that can appear as subquotients of it are of the form S_E,k, where E is the direct product of elementary abelian p-groups. In the special case k = 𝔽₂, the simple Mackey functor S_E,𝔽2 must appear as a filtration factor in 𝔽₂ ⊗_ℤ B for every direct product of elementary abelian p-groups E of odd order.

The noncommuting graph ∇(G) of a non-abelian group G is defined as follows. The vertex set of ∇(G) is G\Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx. It has been conjectured that if P is a finite non-abelian simple group and G is a group such that ∇(P) ≅ ∇(G), then G ≅ P. In the present paper, our aim is to consider this conjecture in the case of finite almost simple groups. In fact, we characterize the projective general linear group PGL(2, q) (q is a prime power), which is also an almost simple group, by its noncommuting graph.

A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A₅, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.

Let be the set of all subgroup normality degrees of a given finite group G and be the union over all , where G ranges over all finite groups. It is shown that and structural results are given for finite groups G with . Also it is proved that G is solvable if or , in which .

Let G be a finite group and nse(G) the set of numbers of elements with the same order in G. In this paper, we prove that if ∣G∣ = ∣S∣ and nse(G) = nse(S), where S is a sporadic simple group, then the finite group G is isomorphic to S.

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.