arXiv subjects

Anton A. Baykalov

Publications and source records attributed to Anton A. Baykalov.

6 recordsLinked to original sources

arXiv2025

Proper partial linear spaces affording imprimitive rank 3 automorphism groups

A partial linear space is a point--line incidence structure such that each line is incident with at least two points and each pair of points is incident with at most one line. It is said to be proper if there exists at least one non-collinear point pair, and at least one line incident with more than two points. The highest degree of symmetry for a proper partial linear space occurs when the automorphism group $G$ is transitive on ordered pairs of collinear points, and on ordered pairs of non-collinear points, that is to say, $G$ is a transitive rank $3$ group on the points. While the primitive rank 3 partial linear spaces are essentially classified, we present the first substantial classification of a family of imprimitive rank $3$ examples. We classify all imprimitive rank $3$ proper partial linear spaces such that the rank $3$ group is semiprimitive. In particular, this includes all partial linear spaces with a rank 3 imprimitive automorphism group that is innately transitive or quasiprimitive. We construct several infinite families of examples and ten individual examples. The examples in the infinite families admit a rank $3$ action of a linear or unitary group, and to our knowledge most of these examples have not appeared before in the literature.

math.GR↗

arXiv2025

Base sizes for finite linear groups with solvable stabilisers

Let $G$ be a transitive permutation group on a finite set with solvable point stabiliser and assume that the solvable radical of $G$ is trivial. In 2010, Vdovin conjectured that the base size of $G$ is at most 5. Burness proved this conjecture in the case of primitive $G$. The problem was reduced by Vdovin in 2012 to the case when $G$ is an almost simple group. Now the problem is further reduced to groups of Lie type through work of Baykalov and Burness. In this paper, we prove the strong form of the conjecture for all almost simple groups with socle isomorphic to $\PSL_n(q),$ and the remaining classical groups will be handled in two forthcoming papers.

math.GR↗

arXiv2024

On algebraic normalisers of maximal tori in simple groups of Lie type

Let $G$ be a finite simple group of Lie type and let $T$ be a maximal torus of $G$. It is well known that if the defining field of $G$ is large enough, then the normaliser of $T$ in $G$ is equal to the algebraic normaliser $N(G,T)$. We identify explicitly all the cases when $N_G(T)$ is not equal to $N(G,T).$

math.GR↗

arXiv2022

Rank three innately transitive permutation groups and related $2$-transitive groups

The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This paper extends classifications of finite primitive and quasiprimitive groups of rank at most $3$ to a classification for the finite innately transitive groups. The new examples comprise three infinite families and three sporadic examples. A necessary step in this classification was the determination of certain configurations in finite almost simple $2$-transitive groups called special pairs.

math.GR↗

arXiv2022

Intersection of conjugate solvable subgroups in finite classical groups

We consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five conjugates of $H$ whose intersection is trivial? This problem is closely related to a conjecture by Babai, Goodman and Pyber (1997) about an upper bound for the index of a normal solvable subgroup in a finite group. In particular, a positive answer to Vdovin's problem yields that if $G$ has a solvable subgroup of index $n$, then it has a solvable normal subgroup of index at most $n^5$. The problem was reduced by Vdovin (2012) to the case when $G$ is an almost simple group. Let $G$ be an almost simple group with socle isomorphic to a simple linear, unitary or symplectic group. For all such groups $G$ we provide a positive answer to Vdovin's problem.

math.GR↗

arXiv2021

Intersection of Solvable Hall subgroups in finite groups

In this paper we show that if S is a simple classical group, a group G is contained in inner-diagonal automorphisms of S and contains S, and H is a solvable Hall subgroup of G, then there exists five conjugates of H, whose intersection is trivial.

math.GR↗