arXiv subjects

A. V. Vasil'ev

Publications and source records attributed to A. V. Vasil'ev.

9 recordsLinked to original sources

arXiv2026

Multivalued Groups and Schur Rings

We study finite involutive multivalued groups associated with Schur rings. Their reduced valency is shown to equal the least common multiple of the sizes of the basic sets of the corresponding ring. We prove that the double-coset group associated with a complemented subgroup of a finite group is isomorphic to a Schur multivalued group over the complement. In particular, every coset multivalued group is a Schur multivalued group. Using affine Schur rings, we construct families of pairwise nonisomorphic commutative multivalued groups over elementary abelian groups. We obtain asymptotic bounds for the sizes of these families and prove that the proportion of double-coset groups in them tends to zero.

math.GR↗

arXiv2024

On finite groups isospectral to groups with abelian Sylow $2$-subgroups

The spectrum of a finite group is the set of orders of its elements. We are concerned with finite groups having the same spectrum as a direct product of nonabelian simple groups with abelian Sylow $2$-subgroups. For every positive integer $k$, we find $k$ nonabelian simple groups with abelian Sylow 2-subgroups such that their direct product is uniquely determined by its spectrum in the class of all finite groups. On the other hand, we prove that there are infinitely many finite groups having the same spectrum as the direct cube of the small Ree group $^2G_2(q)$, $q>3$, or the direct fourth power of the sporadic group $J_1$.

math.GR↗

arXiv2023

On recognition of direct powers of finite simple linear groups by spectrum

The spectrum of a finite group is the set of its element orders. We give an affirmative answer to Problem 20.58(a) from the Kourovka Notebook proving that for every positive integer $k$, the $k$-th direct power of the simple linear group $L_{n}(2)$ is uniquely determined by its spectrum in the class of finite groups provided $n$ is a power of $2$ greater than or equal to $56k^2$.

math.GR↗

arXiv2022

Criterion of nonsolvability of a finite group and recognition of direct squares of simple groups

The spectrum $ω(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if among the prime divisors of the order of a group $G$, there are four different primes such that $ω(G)$ contains all their pairwise products but not a product of any three of these numbers, then $G$ is nonsolvable. Using this result, we show that for $q\geqslant 8$ and $q\neq 32$, the direct square $Sz(q)\times Sz(q)$ of the simple exceptional Suzuki group $Sz(q)$ is uniquely characterized by its spectrum in the class of finite groups, while for $Sz(32)\times Sz(32)$, there are exactly four finite groups with the same spectrum.

math.GR↗

arXiv2021

The $3$-closure of a solvable permutation group is solvable

Let $m$ be a positive integer and let $Ω$ be a finite set. The $m$-closure of $G\leq\operatorname{Sym}(Ω)$ is the largest permutation group on $Ω$ having the same orbits as $G$ in its induced action on the Cartesian product $Ω^m$. The $1$-closure and $2$-closure of a solvable permutation group need not be solvable. We prove that the $m$-closure of a solvable permutation group is always solvable for $m\geq3$.

math.GR↗

arXiv2021

The closures of wreath products in product action

Let $m$ be a positive integer and let $Ω$ be a finite set. The $m$-closure of $G\le$Sym$(Ω)$ is the largest permutation group on $Ω$ having the same orbits as $G$ in its induced action on the Cartesian product $Ω^m$. The exact formula for the $m$-closure of the wreath product in product action is given. As a corollary, a sufficient condition is obtained for this $m$-closure to be included in the wreath product of the $m$-closures of the factors.

math.GR↗

arXiv2018

Simple Groups Whose Prime Graph or Solvable Graph is Split

A graph is split if there is a partition of its vertex set into a clique and an independent set. The present paper is devoted to the splitness of some graphs related to finite simple groups, namely, prime graphs and solvable graphs, and their compact forms. It is proved that the compact form of the prime graph of any finite simple group is split.

math.GR↗

arXiv2018

On recognition of symplectic and orthogonal groups of small dimensions by spectrum

We refer to the set of the orders of elements of a finite group as its spectrum and say that finite groups are isospectral if their spectra coincide. In the paper we determine all finite groups isospectral to the simple groups $S_6(q)$, $O_7(q)$, and $O_8^+(q)$. In particular, we prove that with just four exceptions, every such a finite group is an extension of the initial simple group by a (possibly trivial) field automorphism.

math.GR↗

arXiv2012

Cocliques of maximal size in the prime graph of a finite simple group

In this paper we continue our investgation of the prime graph of a finite simple group started in http://arxiv.org/abs/math/0506294 (the printed version appeared in [1]). We describe all cocliques of maximal size for all finite simple groups and also we correct mistakes and misprints from our previous paper. The list of correction is given in Appendix of the present paper.

math.GR↗