Characterization of Weak EKR Groups and Intersection Densities with Prescribed Point Stabilizers

Abstract: A finite group has the weak Erdos-Ko-Rado property if all of its transitive permutation actions have the EKR property. We characterize this property in terms of normal subgroups and chief factors. More precisely, we introduce a local intersection density and establish a normal-extension criterion which reduces the weak EKR property to difference-set conditions on the elementary abelian chief factors and the linear groups induced on them. For chief factors of rank one the condition is automatically satisfied, and for chief factors of rank two, this condition is equivalent to the induced linear group being intransitive on the one-dimensional subspaces. In the second part of the paper, we solve an open problem by determining the possible intersection densities of transitive permutation groups with a prescribed point stabilizer. We prove that, for every finite group $H$ of order $m\geq 4$ and every integer $n\geq m$, there exists a faithful transitive permutation group with point stabilizer isomorphic to $H$ and intersection density $n/m$.

• The paper demonstrates that finite groups with the weak EKR property satisfy a normal-extension criterion involving conjugate subgroups and automatic solvability.
• The method explicitly involves a local intersection density that matches the global version and involves transitive actions defined by rooted cosets.
• The construction uses a wreath product to establish that any constant $n/m$ can be obtained as a point stabilizer intersection density for transitive actions

This paper, by Frelih, Hujdurović, and Kutnar (2608.18594), makes two contributions to the Erdős–Ko–Rado (EKR) theory of permutation groups. First, it characterizes finite groups with the weak EKR property—those for which every transitive action is EKR—in terms of a normal-extension criterion that reduces to difference-set conditions on elementary abelian chief factors and the linear groups induced on them. Second, it resolves an open problem by showing that every rational number of the form $n/m$, with $m = |H| \geq 4$ and $n \geq m$, occurs as the intersection density of some faithful transitive permutation group whose point stabilizer is isomorphic to $H$.

Background and motivation

For a transitive action of a finite group $G$ on $X$, a subset $\mathcal{F} \subseteq G$ is intersecting if any two of its elements agree on some point; equivalently, $\mathcal{F}$ is an independent set in the derangement graph $\Gamma_G$ (the Cayley graph on $G$ with connection set the derangements). Cosets of point stabilizers are canonical intersecting sets, so the intersection density

$m = |H| \geq 4$0

always satisfies $m = |H| \geq 4$1, with equality exactly when the action has the EKR property. Since the EKR property depends on the action rather than the abstract group, Bardestani and Mallahi-Karai introduced the weak EKR property: $m = |H| \geq 4$2 is weak EKR if $m = |H| \geq 4$3 for every subgroup $m = |H| \geq 4$4. Their result that weak EKR groups are necessarily solvable motivates the central question addressed here: can weak EKR be detected from normal subgroups and chief factors?

A local density and the normal-extension criterion

The key device is a local intersection density. For $m = |H| \geq 4$5, define

$m = |H| \geq 4$6

where $m = |H| \geq 4$7 denotes the union of conjugates of $m = |H| \geq 4$8. The first substantive observation is that this local quantity equals the global one: if $m = |H| \geq 4$9 and $n \geq m$0, then $n \geq m$1. The proof exploits normality directly: since $n \geq m$2, any maximum intersecting set has all pairwise quotients in $n \geq m$3, forcing it to lie in a single coset of $n \geq m$4.

This equality immediately yields a clean normal-extension criterion: $n \geq m$5 has the weak EKR property if and only if (i) $n \geq m$6 has the weak EKR property, and (ii) $n \geq m$7 for every $n \geq m$8. The forward direction uses inheritance under quotients; the converse bounds an arbitrary intersecting set $n \geq m$9 by combining the image bound $H$0 in the quotient with fiberwise bounds $H$1 supplied by the local condition, giving $H$2.

Iterating along a chief series produces the desired structural characterization: $H$3 has the weak EKR property if and only if there exists a chief series $H$4 such that, for each chief factor $H$5, every subgroup $H$6 satisfies the local condition $H$7 in $H$8. Notably, the condition is automatic for chief factors of prime order.

Difference sets and the rank-two criterion

Because weak EKR groups are solvable, every chief factor $H$9 is elementary abelian, say $G$0. Writing $G$1 for the induced linear group, the local condition becomes: for every subspace $G$2,

$G$3

The authors note that this is precisely the equality $G$4 for the parameter $G$5 studied by Matolcsi and Ruzsa in their work on difference sets, specialized to standard sets arising as unions of subspace orbits—so the group-theoretic condition is an instance of a known additive-combinatorial problem, imposed simultaneously over all subspaces and combined across the chief series.

For rank-two chief factors the criterion takes a particularly simple form. Proposition: for $G$6, the condition holds for all $G$7 if and only if $G$8 is intransitive on the set of one-dimensional subspaces of $G$9. If $X$0 is transitive, then $X$1 for a line $X$2, so $X$3 witnesses local density $X$4. Conversely, if $X$5 is intransitive, choose a line $X$6 outside the orbit of $X$7; distinct elements of an admissible $X$8 cannot share a coset of $X$9, whence $\mathcal{F} \subseteq G$0. This yields a complete characterization for solvable groups all of whose chief factors have rank at most 2, and the paper illustrates it with the family $\mathcal{F} \subseteq G$1 for $\mathcal{F} \subseteq G$2: here $\mathcal{F} \subseteq G$3 acts irreducibly but not transitively on lines when $\mathcal{F} \subseteq G$4, so $\mathcal{F} \subseteq G$5 is weak EKR, whereas $\mathcal{F} \subseteq G$6 gives $\mathcal{F} \subseteq G$7, which is not.

Two qualifications are stated plainly. First, the characterization requires solvability via the prior theorem of Bardestani and Mallahi-Karai; no independent proof of solvability emerges from these methods. Second, for chief factors of rank $\mathcal{F} \subseteq G$8 no analogous classification is known—the paper poses as an open problem the characterization of irreducible $\mathcal{F} \subseteq G$9 satisfying the difference-set condition for every subspace $\mathcal{F}$0. The rank-two argument does not extend, because the coset-counting device relies on two distinct lines meeting trivially.

Intersection densities with prescribed point stabilizers

The second part reverses the quantifiers of the weak EKR problem: fix a subgroup $\mathcal{F}$1 and determine which intersection densities arise among transitive groups with point stabilizer isomorphic to $\mathcal{F}$2. Prior work settled $\mathcal{F}$3 (possible densities $\mathcal{F}$4, with $\mathcal{F}$5 excluded) and $\mathcal{F}$6 (all $\mathcal{F}$7, $\mathcal{F}$8). The main theorem completes the picture:

Theorem: For every finite group $\mathcal{F}$9 of order $\Gamma_G$0 and every integer $\Gamma_G$1, there exists a faithful transitive permutation group with point stabilizer isomorphic to $\Gamma_G$2 and intersection density exactly $\Gamma_G$3.

The construction is explicit. Embed $\Gamma_G$4 regularly in $\Gamma_G$5, form the wreath product $\Gamma_G$6, and take the diagonal-like subgroup $\Gamma_G$7. The coset action on $\Gamma_G$8 is faithful (since $\Gamma_G$9) with point stabilizer $G$0.

The upper bound $G$1 proceeds by analyzing supports. Every nonidentity element of a conjugate of $G$2 has support of size two, and an intersecting set normalized to contain the identity consists only of such elements. If all nonidentity elements share a single support, the clique-coclique bound applied inside $G$3 gives $G$4. If multiple supports occur, the support family is pairwise intersecting, hence a star or contained in a triangle; in the star case, at most one element of $G$5 has any given support, yielding $G$6, and the triangle case gives $G$7 (using $G$8). The matching lower bound comes from an explicit intersecting set of size $G$9: fixing a nonidentity $m = |H| \geq 4$00 and using that $m = |H| \geq 4$01 and $m = |H| \geq 4$02 are conjugate in $m = |H| \geq 4$03, the elements supported on $m = |H| \geq 4$04 with entries $m = |H| \geq 4$05 pairwise generate elements lying in conjugates of $m = |H| \geq 4$06.

Two features of this result deserve emphasis. It shows the constraint $m = |H| \geq 4$07 is sharp in scope—every integer above the stabilizer order is realized—so unlike the $m = |H| \geq 4$08 cases, no gaps appear in the spectrum of achievable maximum intersecting set sizes once $m = |H| \geq 4$09. And the construction is uniform: a single wreath-product template depending only on $m = |H| \geq 4$10 and $m = |H| \geq 4$11, requiring no case analysis on the structure of $m = |H| \geq 4$12 beyond its regular embedding. The faithfulness of the action also means these are genuine permutation groups, not merely abstract coset actions.

Limitations and open questions

The characterization in the first part is conditional in two respects. It applies cleanly to chief factors of rank at most 2; for higher-rank factors the difference-set condition $m = |H| \geq 4$13 must be verified ad hoc, and the paper explicitly leaves open its classification for irreducible $m = |H| \geq 4$14 with $m = |H| \geq 4$15. Additionally, the reduction to elementary abelian factors rests on the previously established solvability of weak EKR groups rather than being rederived. In the second part, the realizability theorem covers only stabilizers of order at least 4; the small cases $m = |H| \geq 4$16 and $m = |H| \geq 4$17 exhibit exceptional gaps ($m = |H| \geq 4$18 is unattainable), and the paper does not investigate whether analogous sporadic exclusions might arise for other structures—it simply shows none do for $m = |H| \geq 4$19. The construction also produces highly imprimitive wreath-product actions; whether primitive groups with fixed point stabilizer admit similarly unrestricted densities is not addressed.

Conclusion

The paper delivers a structurally transparent account of the weak EKR property: a local density agreeing with the global one, a normal-extension criterion reducing the property to verifiable conditions on chief factors, and an exact answer in rank two via intransitivity of the induced linear group on projective points—linking the problem to Matolcsi–Ruzsa difference-set parameters. Complementing this, the sharp realizability theorem for prescribed point stabilizers of order at least 4 closes a problem left open for cyclic stabilizers, establishing that the full range $m = |H| \geq 4$20 of intersection densities is attainable uniformly. The principal remaining question posed by the paper is the extension of the rank-two criterion to chief factors of rank three and higher.