Narrow Results

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes, that is, $\mathbb{P}={\cup }_{i\in I}{\sigma }_i$ and ${\sigma }_i\cap {\sigma }_j=\varnothing$ for all $i\ne j$. We say that a finite group $G$ is $\sigma$-soluble if every chief factor $H/K$ of $G$ is a ${\sigma }_i$-group for some $i=i(H/K)\in I$. We give some characterizations of finite $\sigma$-soluble groups.

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.

Let G be a finite group and H a subgroup of G. Recall that H is said to be a TI-subgroup of G if H^g∩ H = 1 or H for each g ∈ G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group G are TI-subgroups, then G is soluble, and all non-nilpotent subgroups of G are normal.

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma (G)=\{{\sigma }_i|{\sigma }_i\cap \pi (G)\ne \varnothing \}$. A set ${\mathscr{H}}$ of subgroups of $G$ is said to be a complete Hall$\sigma$-set of $G$ if every member $\ne 1$ of ${\mathscr{H}}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i\in \sigma$ and ${\mathscr{H}}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma (G)$. A subgroup $A$ of $G$ is said to be $\sigma$-quasinormal or $\sigma$-permutable in $G$ provided there is a complete Hall $\sigma$-set ${\mathscr{H}}$ of $G$ such that $A{H}^x=H^{x}A$ for all $x\in G$ and all $H\in {\mathscr{H}}$.

In this paper, we study the structure of some generalized soluble groups $G$ in which $\sigma$-permutability is a transitive relation in $G$.

Let $G$ be a finite group. A subgroup $H$ of $G$ is called Hall normally embedded in $G$ if $H$ is a Hall subgroup of the normal closure $H^G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that certain subgroups of prime power order are Hall normally embedded in $G$.

The spectrum of a finite group is the set of its element orders. In this paper, we prove that the direct product of two copies of the finite simple sporadic group $J_4$ is uniquely determined by its spectrum in the class of all finite groups.

Let ω be a set of primes with |ω| > 1, and m > n ≥ 0 be integers. It is proved that the lattice of all τ-closed m-multiply ω-saturated formations is not a sublattice of the lattice of all τ-closed n-multiply ω-saturated formations.

Let n > 0 and ω be a non-empty set of primes. It is proved that the lattice of all τ-closed n-multiply ω-composition formations is inductive. It gives a positive answer to a question of Skiba asked in 2001 at the Gomel Algebraic Seminar.

It is proved that every law of the lattice of all τ-closed formations of finite groups is fulfilled in the lattice of all τ-closed n-multiply ω-saturated formations of finite groups, for every subgroup functor τ and every natural number n.

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes. In the 1980s, Thompson conjectured that the equality $N(G)=N(S)$, where $Z(G)=1$ and $S$ is simple, implies the isomorphism $G\simeq S$. In a series of papers of different authors, Thompson’s conjecture was proved. In this paper, we show that in some cases it is possible to omit the conditions $Z(G)=1$ and $S$ is simple and prove a more general result.

The commuting graph ${𝒞}(\mathrm{\Omega },G)$ of a finite group $G$ has vertex set as $\mathrm{\Omega }\subseteq G$, and any two distinct vertices $x,y$ are adjacent if $x$ and $y$ commute with each other. In this paper, we first study the perfect codes of ${𝒞}(\mathrm{\Omega },G)$. We then find the universal adjacency spectra of the join of two regular graphs, join of two regular graphs in which one graph is a union of two regular graphs, and generalized join of regular graphs in terms of adjacency spectra of the constituent graphs and an auxiliary matrix. As a consequence, we obtain the adjacency, Laplacian, signless Laplacian, and Seidel spectra of the above graph operations. As an application of the results obtained, we calculate the adjacency, Laplacian, signless Laplacian, and Seidel spectra of ${𝒞}(G,G)$ for $G\in \{{\mathrm{\text{D}}}_n,{\mathrm{\text{Dic}}}_n,{\mathrm{\text{SD}}}_n\}$, where ${\mathrm{\text{D}}}_n$ is the dihedral group, ${\mathrm{\text{Dic}}}_n$ is the dicyclic group and ${\mathrm{\text{SD}}}_n$ is the semidihedral group. Moreover, we provide the exact value of the spectral radius of the adjacency, Laplacian, signless Laplacian, and Seidel matrix of ${𝒞}(G,G)$ for $G\in \{{\mathrm{\text{D}}}_n,{\mathrm{\text{Dic}}}_n,{\mathrm{\text{SD}}}_n\}$. Some of the theorems published in [F. Ali and Y. Li, The connectivity and the spectral radius of commuting graphs on certain finite groups, Linear Multilinear Algebra 69 (2019) 1–14; T. Cheng, M. Dehmer, F. Emmert-Streib, Y. Li and W. Liu, Properties of commuting graphs over semidihedral groups, Symmetry 13(1) (2021) 103] can be deduced as corollaries from the theorems obtained in this paper.

It is proved that the lattice of totally saturated formations of finite groups is distributive. Thus, we give an affirmative answer to the problem proposed by Shemetkov, Skiba and Guo.

Let $\mathrm{\Gamma }(G)$ denote the Coprime Graph of a finite group $G$. In this paper we study the Laplacian eigenvalues of the Coprime Graph of the finite cyclic group ${\mathbb{Z}}_n$ and the Dihedral group $D_n$ where $n>2$. We find the characteristic polynomial of $\mathrm{\Gamma }({\mathbb{Z}}_n)$ for any $n>2$ and determine the eigenvalues of $\mathrm{\Gamma }({\mathbb{Z}}_n)$ for $n=p^m,pq$ where $p,q$ are primes and $m$ is a positive integer. We characterize the values of $n$ for which algebraic and vertex connectivity of $\mathrm{\Gamma }({\mathbb{Z}}_n)$ are equal. We also discuss about the largest and the second largest eigenvalue of $\mathrm{\Gamma }({\mathbb{Z}}_n)$. Finally, the spectra of $\mathrm{\Gamma }(D_n)$ has been determined for $n=p^m,pq$ where $p,q$ are primes and $m$ is a positive integer.

The subgroup structure of a finite group, under the assumption that its every non-nilpotent maximal subgroup has prime index, is studied in the paper.

Let $G$ be a finite group, $\sigma =\{{\sigma }_i|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$ and $\sigma (G)=\{{\sigma }_i|{\sigma }_i\cap \pi (G)\ne \varnothing \}$. A set ${\mathscr{H}}$ of subgroups of $G$ is said to be a complete Hall$\sigma$-set of $G$ if every non-identity member of ${\mathscr{H}}$ is a Hall ${\sigma }_i$-subgroup of $G$ and ${\mathscr{H}}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma (G)$. A subgroup $H$ of $G$ is said to be $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set ${\mathscr{H}}$ such that $H{A}^x=A^{x}H$ for all $A\in {\mathscr{H}}$ and all $x\in G$. Let $H$ be a subgroup of $G$. $H$ is: $m$-$\sigma$-permutable in $G$ if $H=〈A,B〉$ for some modular subgroup $A$ and $\sigma$-permutable subgroup $B$ of $G$; weakly$m$-$\sigma$-permutable in $G$ if there are an $m$-$\sigma$-permutable subgroup $S$ and a $\sigma$-subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\le S\le H$.

In this paper, we investigate the influence of weakly $m$-$\sigma$-permutable subgroups on the structure of finite groups.

Let $G$ be a group. The enhanced power graph of $G$ is a graph with vertex set $G$ and two distinct vertices $x$ and $y$ are adjacent if there exists $z\in G$ such that $x=z^m$ and $y=z^n$ for some $m,n\in \mathbb{N}$. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other than the identity element. In an article of 2018, Bera and his coauthor characterized all abelian finite groups and nonabelian finite $p$-groups such that their enhanced power graphs are dominatable (see [2]). In addition as an open problem, they suggested characterizing all finite nonabelian groups such that their enhanced power graphs are dominatable. In this paper, we try to answer their question. We prove that the enhanced power graph of finite group $G$ is dominatable if and only if there is a prime number $p$ such that $p\left|\right|Z(G)|$ and the Sylow $p$-subgroups of $G$ are isomorphic to either a cyclic group or a generalized quaternion group.

Let $G$ be a finite group and $H$ a subgroup of $G$. $H$ is said to be $NH$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a Hall subgroup of $G$ and $H\cap T\le H_{\bar{s}G}$, where $H_{\bar{s}G}$ is the largest $s$-semipermutable subgroup of $G$ contained in $H$. In this paper, we give some new characterizations of $p$-nilpotent and supersolvable groups by using $NH$-embedded subgroups. Some known results are generalized.

In this paper, we determine the distance Laplacian spectra of graphs obtained by various graph operations. We obtain the distance Laplacian spectrum of the join of two graphs $G_1$ and $G_2$ in terms of adjacency spectra of $G_1$ and $G_2$. Then we obtain the distance Laplacian spectrum of the join of two graphs in which one of the graphs is the union of two regular graphs. Finally, we obtain the distance Laplacian spectrum of the generalized join of graphs $G_i$, where $1\le i\le n$, in terms of their adjacency spectra. As applications of the results obtained, we have determined the distance Laplacian spectra of some well-known classes of graphs, namely the zero divisor graph of ${\mathbb{Z}}_n$, the commuting and the non-commuting graph of certain finite groups like ${\text{D}}_n$ and ${\mathrm{\text{Dic}}}_n$, and the power graph of various finite groups like ${\mathbb{Z}}_n$, ${\text{D}}_n$ and ${\mathrm{\text{Dic}}}_n$. We show that the zero divisor graph and the power graph of ${\mathbb{Z}}_n$ are distance Laplacian integral for some specific $n$. Moreover, we show that the commuting and the non-commuting graph of ${\text{D}}_n$ and ${\mathrm{\text{Dic}}}_n$ are distance Laplacian integral for all $n\ge 2$.

The main goal of this paper is to provide a group theoretical generalization of the well-known Euler’s totient function. This determines an interesting class of finite groups.

Let $G$ be a finite group and $H$ a subgroup of $G$. Then $H$ is said to be $s$-permutably embedded in $G$ if a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-permutable subgroup of $G$ for every prime dividing the order of $H$. We say that $H$ is nearly$S\mathrm{\Phi }$-embedded in $G$ if $G$ has a normal subgroup $N$ such that $HN$ is $s$-permutable in $G$ and $H\cap N\le \mathrm{\Phi }(H)H_{seG}$, where $H_{seG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutably embedded in $G$. In this paper, we study the properties of the nearly $S\mathrm{\Phi }$-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.