On the intersections of nilpotent subgroups in simple groups

Let $G$ be a finite group and let $H_p$ be a Sylow $p$-subgroup of $G$. A recent conjecture of Lisi and Sabatini asserts the existence of an element $x \in G$ such that $H_p \cap H_p^x$ is inclusion-minimal in the set ${H_p \cap H_p^g ,:, g \in G}$ for every prime $p$. For a simple group $G$, in view of a theorem of Mazurov and Zenkov from 1996, the conjecture implies the existence of an element $x \in G$ with $H_p \cap H_p^x = 1$ for all $p$. In turn, this statement implies a conjecture of Vdovin from 2002, which asserts that if $G$ is simple and $H$ is a nilpotent subgroup, then $H \cap H^x = 1$ for some $x \in G$. In this paper, we adopt a probabilistic approach to prove the Lisi-Sabatini conjecture for all non-alternating simple groups. By combining this with earlier work of Kurmazov on nilpotent subgroups of alternating groups, we complete the proof of Vdovin’s conjecture. Moreover, by combining our proof with earlier work of Zenkov on alternating groups, we are able to establish a stronger form of Vdovin’s conjecture: if $G$ is simple and $A,B$ are nilpotent subgroups, then $A \cap B^x = 1$ for some $x \in G$. To obtain these results, we study the probability that a random pair of Sylow $p$-subgroups in a simple group of Lie type intersect trivially, complementing recent work of Diaconis et al. and Eberhard on symmetric and alternating groups.

💡 Research Summary

The paper addresses a recent conjecture of Lisi and Sabatini which predicts, for any finite group G, the existence of an element x such that for each prime divisor p of |G| the intersection of a Sylow p‑subgroup Hₚ with its conjugate Hₚˣ is inclusion‑minimal among all such intersections. When G is simple, Mazurov and Zenkov’s 1996 theorem forces this minimal intersection to be trivial, so the conjecture reduces to the statement that there is an x with Hₚ∩Hₚˣ=1 for every p. This “global” version (Conjecture 6) immediately implies Vdovin’s 2002 conjecture that any nilpotent subgroup H of a simple group meets a conjugate trivially, and it also yields stronger statements about bases and subgroup depth.

The authors prove Conjecture 6 for all non‑alternating finite simple groups by a probabilistic method. For each prime p they define Qₚ(G) as the proportion of elements x∈G for which Hₚ∩Hₚˣ≠1. The overall failure probability Q(G) that some prime fails is bounded by the sum of the Qₚ(G)’s, so it suffices to show Σₚ Qₚ(G)<1. To bound each Qₚ(G) they view G acting transitively on the coset space G/Hₚ; then Qₚ(G) equals the probability that two random points do not form a base. Using a refinement of a bound due to Liebeck and Shalev, they obtain

bQₚ(G)=∑_{i=1}^k (|x_i^G∩Hₚ|/|x_i^G|)²,

where the x_i run over representatives of the conjugacy classes of elements of order p. Thus the problem reduces to estimating the size of the intersection of a p‑element class with a Sylow subgroup. For most classes the trivial bound |x_i^G∩Hₚ|<|Hₚ| already yields a sufficiently small contribution. When the class is exceptionally small, the authors embed Hₚ into a larger, more tractable subgroup L and bound |x_i^G∩L| instead.

A special treatment is required for the defining characteristic r of a group of Lie type. In this case H_r is the unipotent radical of a Borel subgroup, and by considering the opposite Borel one sees directly that H_r∩H_rˣ=1 for some x, giving the strong bound Q_r(G)≤1−|H_r|·|N_G(H_r)|/|G|. Consequently the product over the remaining primes can be made arbitrarily small, establishing Σₚ Qₚ(G)<1 for all such groups. The only exceptional simple group where the general argument does not apply is U₄(2)≅PSp₄(3); the authors verify the conjecture for this group by explicit computation.

The main results are:

• Theorem A – Conjecture 6 holds for every non‑alternating finite simple group.
• Corollary B – Vdovin’s conjecture (nilpotent subgroups intersect trivially with a suitable conjugate) holds for all finite simple groups.
• Corollary C – For any non‑trivial nilpotent subgroup H of a simple group, the base size b(G,H)=2 and the subgroup depth d_G(H)=3.
• Theorem D – For simple groups of Lie type, the product β(G)=∏_{p≠r} Qₚ(G) tends to 0 as |G|→∞; in particular β(G)→0 for L₂(q) as q→∞.
• Corollary E – For any infinite sequence (G_i,p_i) with |G_i|→∞, either Q_{p_i}(G_i)→0 or the sequence falls into one of three explicit families: alternating groups with p_i=2, L₂(q) with q a Mersenne prime and p_i=2, or Lie type groups in characteristic p_i.
• Theorem F and Corollary G – For any two nilpotent subgroups A,B of a simple group, there exists x with A∩Bˣ=1; equivalently, every pair of nilpotent subgroups is regular.

The paper also discusses connections to permutation group bases, subgroup depth, and p‑block theory, showing that the probabilistic proof is independent of the earlier representation‑theoretic arguments of Mazurov‑Zenkov. By providing explicit upper bounds for the probabilities Qₚ(G) across all families of simple groups, the authors give a uniform, quantitative treatment of the intersection problem, opening the way for further applications of probabilistic methods in the study of subgroup intersections in finite groups.