arXiv subjects

Nanying Yang

Publications and source records attributed to Nanying Yang.

12 recordsLinked to original sources

arXiv2024

On groups whose conjugacy class sizes are not divisible by each other

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes excluding~$1$. Let us define a directed graph $\Gamma(G)$, the set of vertices of this graph is $N(G)$ and the vertices $x$ and $y$ are connected by a directed edge from $x$ to $y$ if $x$ divides $y$ and $N(G)$ does not contain a number $z$ different from $x$ and $y$ such that $x$ divides $z$ and $z$ divides $y$. We will call the graph $\Gamma(G)$ the conjugate graph of the group $G$. In this work, we will study finite groups whose conjugate graph is a set of points.

math.GR↗

arXiv2024

On combinatorial properties of Gruenberg--Kegel graphs of finite groups

If $G$ is a finite group, then the spectrum $\omega(G)$ is the set of all element orders of $G$. The prime spectrum $\pi(G)$ is the set of all primes belonging to $\omega(G)$. A simple graph $\Gamma(G)$ whose vertex set is $\pi(G)$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if $rs \in \omega(G)$ is called the Gruenberg-Kegel graph or the prime graph of $G$. In this paper, we prove that if $G$ is a group of even order, then the set of vertices which are non-adjacent to $2$ in $\Gamma(G)$ form a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg-Kegel graph of a finite group. Besides this, we prove that a complete bipartite graph with each part of size at least $3$ can not be isomorphic to the Gruenberg-Kegel graph of a finite group.

math.GR↗

arXiv2021

Finite groups isospectral to simple groups

The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.

math.GR↗

arXiv2021

On the sharp Baer--Suzuki theorem for $π$-radicals: sporadic groups

Let $π$ be a proper subset of the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m = r$ if $r = 2$ or $3$ and $m = r-1$ if $r \geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in the $π$-radical $\mathrm{O}_π(G)$ of $G$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to sporadic or alternating groups.

math.GR↗

arXiv2021

On the sharp Baer--Suzuki theorem for the $π$-radical

Let $π$ be a set of primes such that $|π|\geqslant 2$ and $π$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $π$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the following conjecture: a conjugacy class $D$ of a finite group $G$ is contained in $Oπ(G)$ if and only if every $m$ elements of $D$ generate a $π$-subgroup. We confirm this conjecture for each group $G$ whose nonabelian composition factors are isomorphic to alternating, linear and unitary simple groups.

math.GR↗

arXiv2020

The minimal polynomials of powers of cycles in the ordinary representations of symmetric and alternating groups

Denote the alternating and symmetric groups of degree $n$ by $A_n$ and $S_n$ respectively. Consider a permutation $σ\in S_n$ all of whose nontrivial cycles are of the same length. We find the minimal polynomials of $σ$ in the ordinary irreducible representations of $A_n$ and $S_n$.

math.GR↗

arXiv2019

Baer--Suzuki theorem for the $π$-radical

In the paper we prove (modulo the classification of finite simple groups) an analogue of the famous Baer-Suzuki theorem for the $π$-radical of a finite group, where $π$ is a set of primes

math.GR↗

arXiv2019

On the nilpotency of the solvable radical of a finite group isospectral to a simple group

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.

math.GR↗

arXiv2017

On injectors of Hartley set of a finite group

Let G be a group and H be a Hartley set of G. In this paper, we prove the existence and conjugacy of H-injectors of G and describe the structure of the injectors. As application, some known results are directly followed.

math.GR↗

arXiv2017

On the problem of existence and conjugacy of injectors of generalized $π$-soluble groups

In this paper, we prove the existence and conjugacy of injectors of a generalized $π$-soluble groups for the Hartley class defined by a invariable Hartley function, and give a description of the structure of the injectors.

math.GR↗

arXiv2017

On F-injectors of Fitting set of a finite group

Let $G$ be some generalized $π$-soluble groups and ${\cal F}$ be a Fitting set of $G$. In this paper, we prove the existence and conjugacy of ${\cal F}$-injectors of $G$, and give a description of the structure of the injectors.

math.GR↗

arXiv2017

Finite groups with permutable Hall subgroups

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 ${\cal H}$.

math.GR↗