arXiv subjects

Saveliy V. Skresanov

Publications and source records attributed to Saveliy V. Skresanov.

At least 19 recordsLinked to original sources

arXiv2026

Polynomial-time isomorphism test for solvable groups with abelian Sylow subgroups

The group isomorphism problem in computational complexity asks whether two finite groups given by their Cayley tables are isomorphic or not. Although polynomial-time isomorphism tests exist for many specific types of groups, no general polynomial-time algorithm is known, classes of solvable and nilpotent groups being the main obstacles. In 2012 Babai and Qiao gave a polynomial-time isomorphism test for the class of solvable groups admitting normal series with abelian Sylow factors. We generalize their result and give a polynomial-time isomorphism test for solvable A-groups, i.e. solvable groups with abelian Sylow subgroups. The algorithm heavily relies both on the computational methods developed by Babai and Qiao, and structural properties of A-groups.

math.GR↗

arXiv2026

Polynomial-time isomorphism test for $k$-generated extensions of abelian groups

The group isomorphism problem asks whether two finite groups given by their Cayley tables are isomorphic or not. Although there are polynomial-time algorithms for some specific group classes, the best known algorithm for testing isomorphism of arbitrary groups of order $ n $ has time complexity $ n^{O(\log n)} $. We consider the group isomorphism problem for some extensions of abelian groups by $ k $-generated groups for bounded $ k $. In particular, we prove that one can test isomorphism of abelian-by-cyclic extensions in polynomial time, generalizing a 2009 result of Le Gall for coprime extensions. As another application, we give a polynomial-time isomorphism test for abelian-by-simple group extensions, generalizing a 2017 result of Grochow and Qiao for central extensions. The main novelty of the proof is a polynomial-time algorithm for computing the unit group of a finite ring, which might be of independent interest.

math.GR↗

arXiv2025

On finite groups with exactly one noncommutator

An element $x$ of a group $G$ is a commutator if it can be expressed in the form $x = a^{-1}b^{-1}ab$ for some $a, b \in G$. In 2010 MacHale posed the following problem in the Kourovka notebook: does there exist a finite group $G$, with $|G| > 2$, such that there is exactly one element of $G$ which is not a commutator? We answer this question in the affirmative and provide an infinite series of such groups, the smallest group in our construction having size $16609443840$.

math.GR↗

arXiv2025

Rota-Baxter operators on compact simple Lie groups and algebras

A Rota-Baxter operator on a Lie group $ G $ is a smooth map $ B : G \to G $ such that $ B(g)B(h) = B(gB(g)hB(g)^{-1}) $ for all $ g, h \in G $. This concept was introduced in 2021 by Guo, Lang and Sheng as a Lie group analogue of Rota-Baxter operators of weight 1 on Lie algebras. We show that the only Rota-Baxter operators on compact simple Lie groups are the trivial map and the inverse map. A similar description for Rota-Baxter operators of weight 1 on compact simple Lie algebras is provided.

math.GR↗

arXiv2025

Reduction of the group isomorphism problem to the group automorphism problem

It is well known that the graph isomorphism problem is polynomial-time reducible to the graph automorphism problem (in fact these two problems are polynomial-time equivalent). We show that, analogously, the group isomorphism problem is polynomial-time reducible to the group automorphism problem. Reductions to other relevant problems like automorphism counting are also given.

cs.CC↗

arXiv2025

Random unipotent Sylow subgroups of groups of Lie type of bounded rank

In 2001 Liebeck and Pyber showed that a finite simple group of Lie type is a product of $ 25 $ carefully chosen unipotent Sylow subgroups. Later, in a series of works it was shown that $ 4 $ unipotent Sylow subgroups suffice. We prove that if the rank of a finite simple group of Lie type $ G $ is bounded, then $ G $ is a product of $ 11 $ random unipotent Sylow subgroups with probability tending to $ 1 $ as $ |G| $ tends to infinity. An application of the result to finite linear groups is given. The proofs do not depend on the classification of finite simple groups.

math.GR↗

arXiv2024

Expanders and growth of normal subsets in finite simple groups of Lie type

We show that some classical results on expander graphs imply growth results on normal subsets in finite simple groups. As one application, it is shown that given a nontrivial normal subset $ A $ of a finite simple group $ G $ of Lie type of bounded rank, we either have $ G \setminus \{ 1 \} \subseteq A^2 $ or $ |A^2| \geq |A|^{1+\epsilon} $, for $ \epsilon > 0 $. This improves a result of Gill, Pyber, Short and Szab\'o, and partially resolves a question of Pyber from the Kourovka notebook. We also propose a variant of Gowers' trick for two subsets, and give applications to products of large subsets in groups of Lie type, improving some results of Larsen, Shalev and Tiep.

math.GR↗

arXiv2024

Closures of permutation groups with restricted nonabelian composition factors

Given a permutation group $G$ on a finite set $\Omega$, let $G^{(k)}$ denote the $k$-closure of $G$, that is, the largest permutation group on $\Omega$ having the same orbits in the induced action on $\Omega^k$ as $G$. Recall that a group is $\mathrm{Alt}(d)$-free if it does not contain a section isomorphic to the alternating group of degree $d$. Motivated by some problems in computational group theory, we prove that the $k$-closure of an $\mathrm{Alt}(d)$-free group is again $\mathrm{Alt}(d)$-free for $k \geq 4$ and $d \geq 25$.

math.GR↗

arXiv2024

A generalization of the Brauer-Fowler theorem

The famous Brauer-Fowler theorem states that the order of a finite simple group can be bounded in terms of the order of the centralizer of an involution. Using the classification of finite simple groups, we generalize this theorem and prove that if a simple locally finite group has an involution which commutes with at most $n$ involutions, then the group is finite and its order is bounded in terms of $n$ only. This answers a question of Strunkov from the Kourovka notebook.

math.GR↗

arXiv2023

On directed and undirected diameters of vertex-transitive graphs

A directed diameter of a directed graph is the maximum possible distance between a pair of vertices, where paths must respect edge orientations, while undirected diameter is the diameter of the undirected graph obtained by symmetrizing the edges. In 2006 Babai proved that for a connected directed Cayley graph on $n$ vertices the directed diameter is bounded above by a polynomial in undirected diameter and $\log n$. Moreover, Babai conjectured that a similar bound holds for vertex-transitive graphs. We prove this conjecture of Babai, in fact, it follows from a more general bound for connected relations of homogeneous coherent configurations. The main novelty of the proof is a generalization of Ruzsa's triangle inequality from additive combinatorics to the setting of graphs.

math.CO↗

arXiv2023

On the automorphism group of a distance-regular graph

The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on $n$ vertices of diameter greater than two is at least $n/C$ for some universal constant $C > 0$, unless the graph is a Johnson or Hamming graph. We prove that the motion of a distance-regular graph of diameter $d \geq 3$ on $n$ vertices is at least $Cn/(\log n)^6$ for some universal constant $C > 0$, unless it is a Johnson, a Hamming or a crown graph. This follows using an improvement of an earlier result by Kivva who gave a lower bound on motion of the form $n/c_d$, where $c_d$ depends exponentially on $d$. As a corollary we derive a quasipolynomial upper bound for the automorphism group of a primitive distance-regular graph acting edge-transitively on the graph and on its distance-2 graph. The proofs use elementary combinatorial arguments and do not depend on the classification of finite simple groups.

math.CO↗

arXiv2023

On the orders of composition factors in completely reducible groups

We obtain an asymptotic upper bound for the product of the $p$-parts of the orders of certain composition factors of a finite group acting completely reducibly and faithfully on a finite vector space of order divisible by a prime $p$. An application is given for the diameter of a nondiagonal orbital graph of an affine primitive permutation group.

math.GR↗

arXiv2023

On a polynomial bound for the orbital diameter of primitive affine groups

Let $ VG $ be a finite primitive affine permutation group, where $ V $ is a vector space of dimension $ d $ over the prime field $ \mathbb{F}_p $ and $ G $ is an irreducible linear group on $ V $. We prove that if $ p $ divides $ |G| $, then the diameters of all nondiagonal orbital graphs of $ VG $ are at most $ 9d^3 $. This improves an earlier exponential bound by A. Mar\'oti and the author.

math.GR↗

arXiv2022

Bounds for the diameters of orbital graphs of affine groups

General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer $ H \leq \mathrm{GL}(V) $, where the vector space $ V $ has dimension $ d $ over the prime field, can be bounded in terms of $ d $ and $ \log |V| / \log |H| $ only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.

math.GR↗

arXiv2022

Two-closure of rank 3 groups in polynomial time

A finite permutation group $G$ on $\Omega$ is called a rank 3 group if it has precisely three orbits in its induced action on $\Omega \times \Omega$. The largest permutation group on $\Omega$ having the same orbits as $G$ on $\Omega \times \Omega$ is called the 2-closure of $G$. We construct a polynomial-time algorithm which given generators of a rank 3 group computes generators of its 2-closure.

math.GR↗

arXiv2021

The probabilistic Weisfeiler-Leman algorithm

A probabilistic version of the Weisfeiler-Leman algorithm for computing the coherent closure of a colored graph is suggested. The algorithm is Monte Carlo and runs in time $ O(n^{1+\omega}\log^2 n) $, where $ n $ is the number of vertices of the graph and $ \omega < 2.273 $ is the matrix multiplication exponent.

cs.CC↗

arXiv2020

On 2-closures of rank 3 groups

A permutation group $G$ on $\Omega$ is called a rank 3 group if it has precisely three orbits in its induced action on $\Omega \times \Omega$. The largest permutation group on $\Omega$ having the same orbits as $G$ on $\Omega \times \Omega$ is called the 2-closure of $G$. A description of 2-closures of rank 3 groups is given. As a special case, it is proved that 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional.

math.GR↗

arXiv2020

On element orders in covers of $L_4(q)$ and $U_4(q)$

Suppose that $L$ is one of the finite simple groups $PSL_4(q)$ or $PSU_4(q)$ and $L$ acts on a vector space $W$ over a field whose characteristic divides $q$. We prove that the natural semidirect product of $W$ and $L$ contains an element whose order differs from the order of any element of $L$, thus answering questions 14.60 and 17.73 (a) of the Kourovka Notebook.

math.GR↗