arXiv subjects
Publications and source records attributed to Maria A. Grechkoseeva.
For a finite group $G$, let $ω(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $ω(H)=ω(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if it is finite, then all finite groups $H$ with $ω(H)=ω(G)$ are listed. We complete the solution of the recognition problem for all finite simple groups.
For a finite group $G$, let $ω(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $ω(H)=ω(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if $h(G)$ is finite, then all finite groups $H$ with $ω(H)=ω(G)$ are described. We complete the solution of the recognition problem for the finite simple linear and unitary groups.
Let $L$ be one of the finite simple classical groups $L_8(q)$, $U_8(q)$, $O_{10}^+(q)$, $O_{10}^-(q)$ or $O_{12}^+(q)$, with $q$ odd. We prove that every finite group having the same set of element orders as $L$ is an almost simple group with socle isomorphic to $L$. This completes the study of the recognition-by-spectrum problem for simple classical groups whose prime graph independence number is equal to $4$.
We give an explicit description of the set of element orders for every almost simple group with socle $E_7(q)$.
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.
We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of $G$ by the solvable radical of $G$ is an almost simple group. The main goal of this paper is prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when $G$ is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime graph of $G$, where the prime graph of $G$ is the graph whose vertices are the prime numbers dividing the order of $G$ and two such numbers $r$ and $s$ are adjacent if and only if $r\neq s$ and $G$ has an element of order $rs$. Namely, we establish some sufficient conditions for the prime graph of such a group to have a vertex adjacent to all other vertices. Besides proving the main result, this allows us to refine a recent result by P. Cameron and N. Maslova concerning finite groups almost recognizable by prime graph.