Vdovin's Conjecture in Finite Group Theory

• Vdovin's Conjecture is a family of finite-group-theoretic conjectures that examines intersections of conjugate subgroups with strong internal restrictions, providing sharp bounds on base size.
• The conjecture comprises distinct formulations—the five-conjugate, nilpotent-subgroup, and torus-normaliser versions—each addressing trivial or small intersections under specific group conditions.
• Recent advances for almost simple groups, particularly classical families like PSL_n(q), confirm optimal bounds (e.g., base size 5) by blending structural decompositions and probabilistic class-size estimates.

Vdovin’s Conjecture is not a single universally fixed statement in the literature, but a family of finite-group-theoretic conjectures attributed to Vdovin and centered on intersections of conjugate subgroups. The common theme is that subgroups with strong internal restrictions—solvable subgroups, nilpotent subgroups, or normalisers of maximal tori—should admit a small number of conjugates with very small intersection, often trivial intersection. In permutation-group language, these statements are naturally expressed in terms of base size for a coset action. Within the cited literature, the nilpotent-subgroup version for finite simple groups is proved in full, the five-conjugate solvable-subgroup version is established in major almost simple families, and the maximal-torus-normaliser version is resolved in a stronger modified form with a short exceptional list (Baykalov, 2017, Burness et al., 2022, Baykalov, 2024, Burness et al., 5 Aug 2025).

1. Scope of the term in the literature

In the cited papers, “Vdovin’s conjecture” refers to several distinct statements rather than a single canonical formulation. The three principal versions are summarized below.

Version	Statement	Setting
Five-conjugate conjecture	Five conjugates of a solvable subgroup should have trivial intersection	Finite groups with no nontrivial solvable normal subgroups
Nilpotent-subgroup conjecture	If $G$ is simple and $H\le G$ is nilpotent, then $H\cap H^x=1$ for some $x\in G$	Finite non-abelian simple groups
Torus-normaliser conjecture	If $G\neq L_3(2)$, then $N\cap N^x$ is a $p$-group for some $x\in G$	Finite simple groups of Lie type, with $N$ the normaliser of a maximal torus

The five-conjugate formulation appears as Kourovka Notebook Problem 17.41(b): if $H$ is a solvable subgroup of a finite group $H\le G$0 with no nontrivial solvable normal subgroups, must there always exist five conjugates of $H\le G$1 whose intersection is trivial? The nilpotent formulation is stated as a 2002 conjecture: if $H\le G$2 is a finite simple group and $H\le G$3 is nilpotent, then there exists $H\le G$4 such that $H\le G$5. The Lie-type torus-normaliser version asks, for $H\le G$6 and $H\le G$7, whether $H\le G$8 is a $H\le G$9-group for some $H\cap H^x=1$0, with the explicit exception $H\cap H^x=1$1 in the original statement (Baykalov, 2017, Burness et al., 2022, Burness et al., 5 Aug 2025).

A separate differential-algebra paper, “The Dimension Conjecture Implies The Jacobi Bound Conjecture,” explicitly states that Vdovin’s Conjecture is not mentioned there at all and that no implication involving a conjecture attributed to Vdovin is established. This is relevant because it excludes a direct connection between Vdovin’s Conjecture and the Dimension/Jacobi-bound framework (Dupuy et al., 18 Mar 2026).

2. The five-conjugate formulation and its base-size interpretation

The 2010 conjecture is most naturally phrased in terms of base size. If $H\cap H^x=1$2 acts on the cosets of a solvable subgroup $H\cap H^x=1$3, and

$H\cap H^x=1$4

then the statement that five conjugates of $H\cap H^x=1$5 have trivial intersection is equivalent to

$H\cap H^x=1$6

where $H\cap H^x=1$7 is the base size of the faithful action of $H\cap H^x=1$8 on $H\cap H^x=1$9. In the transitive permutation-group formulation used later, if the point stabiliser $x\in G$0 is solvable and the solvable radical of $x\in G$1 is trivial, then Vdovin’s conjecture becomes the assertion

$x\in G$2

This formulation is connected to the Babai–Goodman–Pyber conjecture: if a finite group $x\in G$3 has a solvable subgroup of index $x\in G$4, then a positive answer would imply that $x\in G$5 has a solvable normal subgroup of index at most $x\in G$6 (Baykalov, 2017, Baykalov, 2024).

The same literature emphasizes that the bound $x\in G$7 is optimal. One explicit example is

$x\in G$8

for which $x\in G$9. Thus the conjecture is not merely qualitative; it proposes a sharp universal upper bound in the radical-free solvable-stabiliser setting (Baykalov, 2024).

A further strengthening used in proofs is the regular-orbit condition $G\neq L_3(2)$0. Both the 2017 and 2024 works exploit the lemma

$G\neq L_3(2)$1

so the strong form can often be deduced once a four-point base is available. This regular-orbit viewpoint is especially useful in reductions to almost simple groups (Baykalov, 2017, Baykalov, 2024).

3. Progress for almost simple linear, unitary, and symplectic groups

A major step was the verification of the five-conjugate conjecture for almost simple groups with classical socle of type $G\neq L_3(2)$2 or $G\neq L_3(2)$3. For a finite almost simple group $G\neq L_3(2)$4 with socle

$G\neq L_3(2)$5

the 2017 paper proves that if $G\neq L_3(2)$6 is solvable, then

$G\neq L_3(2)$7

and hence

$G\neq L_3(2)$8

This gives a positive answer to Vdovin’s problem for those almost simple groups (Baykalov, 2017).

The proof architecture is structural rather than purely probabilistic. It proceeds by reducing to maximal solvable subgroups and then treating the linear, unitary, and symplectic families separately. The tools highlighted in the paper include primitive and quasi-primitive solvable linear groups, Aschbacher-type decompositions, Singer cycles and their normalizers, fixed point ratio estimates, and explicit conjugating matrices that force successive intersections down to diagonal subgroups, scalar subgroups, or the center. For irreducible maximal solvable subgroups of $G\neq L_3(2)$9, the paper proves

$N\cap N^x$0

except for $N\cap N^x$1, where the exceptional subgroup $N\cap N^x$2 has base size $N\cap N^x$3. For irreducible maximal solvable subgroups of $N\cap N^x$4, it proves

$N\cap N^x$5

except for a small low-rank situation $N\cap N^x$6, $N\cap N^x$7 (Baykalov, 2017).

The 2024 paper then establishes the strong form of the conjecture for all almost simple groups with socle $N\cap N^x$8. Its main theorem states that if $N\cap N^x$9 is transitive, almost simple, its socle is isomorphic to $p$0, and the point stabiliser $p$1 is solvable, then

$p$2

hence

$p$3

It further proves that for $p$4, if $p$5 is a maximal solvable subgroup of

$p$6

not contained in $p$7, then either $p$8, or $p$9, $x\in G$0, $x\in G$1 is the normaliser of the stabiliser of a $x\in G$2-dimensional subspace, and

$x\in G$3

This places the full linear case within the affirmative range of the conjecture (Baykalov, 2024).

4. The nilpotent-subgroup conjecture for finite simple groups

A different and older statement, attributed to Vdovin in 2002, concerns nilpotent rather than solvable subgroups and restricts to finite simple groups. Its formulation is: $x\in G$4 When $x\in G$5 is core-free and $x\in G$6 acts on $x\in G$7, this is equivalent to the assertion that the corresponding base size is $x\in G$8 (Burness et al., 5 Aug 2025).

The 2025 paper situates this conjecture within a stronger Sylow-subgroup framework. Let $x\in G$9, and for each $N$0 let $N$1 be a Sylow $N$2-subgroup. The Lisi–Sabatini conjecture asks for an element $N$3 such that each $N$4 is inclusion-minimal in the set $N$5. For a finite simple group, a theorem of Mazurov and Zenkov implies that for every prime $N$6 and every Sylow $N$7-subgroup $N$8, there exists $N$9 with

$H$0

Since every nilpotent subgroup is the direct product of its Sylow subgroups, the simultaneous trivial-intersection statement for all primes implies Vdovin’s conjecture (Burness et al., 5 Aug 2025).

The paper proves precisely that simultaneous statement for all finite non-alternating simple groups: $H$1 Combining this with Kurmazov’s earlier result for alternating groups yields the corollary that Vdovin’s conjecture is true for all finite simple groups. The same paper also proves a stronger theorem for groups of Lie type: $H$2 and, using earlier work of Zenkov on alternating and sporadic groups, extends this stronger two-subgroup statement to all finite simple groups. As consequences, every nontrivial nilpotent subgroup $H$3 of a simple group has base size $H$4 and subgroup depth $H$5 (Burness et al., 5 Aug 2025).

5. Normalisers of maximal tori in finite simple groups of Lie type

Another conjecture due to Vdovin concerns torus normalisers in finite simple groups of Lie type. Here

$H$6

is a finite simple group of Lie type over $H$7 of characteristic $H$8, $H$9 is an $H\le G$00-stable maximal torus, $H\le G$01, and

$H\le G$02

The conjecture is stated as follows: if $H\le G$03, then $H\le G$04 is a $H\le G$05-group for some $H\le G$06. The exception $H\le G$07 is genuine: if $H\le G$08 is the normaliser of a Singer cycle, then $H\le G$09 for all $H\le G$10 (Burness et al., 2022).

This problem is recast in terms of the action of $H\le G$11 on

$H\le G$12

The base size $H\le G$13 is the minimum size of a subset of $H\le G$14 whose pointwise stabiliser in $H\le G$15 is trivial, equivalently the smallest number of conjugates of $H\le G$16 whose intersection is trivial. In particular,

$H\le G$17

The paper further notes that if

$H\le G$18

then

$H\le G$19

Thus the conjecture becomes a base-size problem except when a nontrivial $H\le G$20-part in the Weyl-group component obstructs this equivalence (Burness et al., 2022).

The main theorem classifies all cases where the base size is not $H\le G$21: either

$H\le G$22

or $H\le G$23 appears in Table 1 of the paper. The exceptional pairs listed there are

$H\le G$24

$H\le G$25

$H\le G$26

and in these cases $H\le G$27. From this, the authors derive a modified form of the conjecture: there exists $H\le G$28 such that $H\le G$29 is a $H\le G$30-group if and only if $H\le G$31 is not one of

$H\le G$32

Accordingly, the original exception $H\le G$33 must be supplemented by two additional small unitary exceptions in the modified theorem (Burness et al., 2022).

6. Methods, structural themes, and current status

Across these formulations, base size acts as the common organizing invariant. In the torus-normaliser problem, the probabilistic analysis is built on

$H\le G$34

with the conclusion that $H\le G$35 forces $H\le G$36. In the nilpotent-subgroup problem, one defines

$H\le G$37

and if

$H\le G$38

then there exists a single $H\le G$39 such that $H\le G$40 for all $H\le G$41. The same paper strengthens this via

$H\le G$42

reducing the argument to class-size estimates and counts of prime-order elements in Sylow subgroups (Burness et al., 2022, Burness et al., 5 Aug 2025).

The five-conjugate program for almost simple classical groups combines this probabilistic framework with explicit subgroup structure. The cited proofs use primitive and quasi-primitive solvable linear groups, Aschbacher-type decompositions, block-upper-triangular forms, Singer cycles and their normalizers, fixed point ratio estimates, explicit conjugating matrices, and computational verification in small cases with $H\le G$43, $H\le G$44, or MAGMA. In the 2024 linear-group paper, the function

$H\le G$45

and the invariant

$H\le G$46

are used to convert bounds on conjugacy-class sizes into base-size bounds. This is characteristic of the broader methodology: structural decomposition reduces to irreducible or primitive configurations, and probabilistic class-sum inequalities eliminate most remaining cases (Baykalov, 2017, Baykalov, 2024).

The present status is therefore differentiated by formulation. The 2002 nilpotent-subgroup conjecture for finite simple groups is proved, indeed in the stronger form that for any two nilpotent subgroups $H\le G$47 of a finite simple group there exists $H\le G$48 with $H\le G$49. The 2010 five-conjugate conjecture is established for the classical almost simple families explicitly treated in the cited papers, including the full linear case $H\le G$50, and earlier positive results for unitary and symplectic socles. The maximal-torus-normaliser conjecture is essentially correct but only after the modification that adds $H\le G$51 and $H\le G$52 to the original exception $H\le G$53. A recurrent source of confusion is to conflate these three statements under one name; the cited literature instead shows that “Vdovin’s Conjecture” is a label for several related, but mathematically distinct, intersection problems in finite group theory (Burness et al., 2022, Baykalov, 2024, Burness et al., 5 Aug 2025).