Graph-restrictiveness of semiprimitive groups with a regular normal nilpotent subgroup: the exceptional cases where 2 or 3 divides the degree (the Potočnik–Spiga–Verret conjecture)

The Potočnik–Spiga–Verret (PSV) conjecture predicts that a finite permutation group is graph-restrictive — meaning vertex stabilisers in all locally-L arc-transitive graphs are bounded by a constant depending only on the local action L — if and only if it is semiprimitive. Giudici and Morgan (2015) proved the conjecture for semiprimitive groups admitting a regular normal nilpotent subgroup R whenever the degree |R| is coprime to 6, and showed that any failure must involve a nontrivial edge-kernel that is a 2- or 3-group, forcing normal sections of the local action isomorphic to direct products of copies of Sym(3) or of Alt(4). Whether groups in these exceptional configurations — including the explicit families of Frobenius groups P⋊C2 with P a non-cyclic abelian 3-group with Φ(P)≠1, the degree-4^m groups W⋊GL(V) with V=F_4 or F_2^2 and m>1, and the extraspecial towers beginning with Q_8 — are graph-restrictive has remained undecided since 2015. The problem asks for a complete determination: either a proof that every semiprimitive group with a regular normal nilpotent subgroup is graph-restrictive, or a single counterexample, which would refute the PSV conjecture outright. As of late 2025 the question is open. Among these explicit families the smallest undecided cases are the degree-16 groups W⋊GL(V) with W=V^2 (Example 1.5) and the degree-27 group with P≅C9×C3 (Example 1.4); the cyclic instances of Example 1.4 are dihedral groups of odd degree — including the group of order 18 acting on 9 points — and are already graph-restrictive, with c(L)=16, by Example 11 of Potočnik–Spiga–Verret.

Background

A finite graph Γ is G-arc-transitive, for a subgroup G of its automorphism group, when G is transitive on the arcs (ordered pairs of adjacent vertices) of Γ. For a vertex x, the vertex stabiliser G_x induces a permutation group $G_x^{\Gamma(x)}$ on the neighbourhood $\Gamma(x)$, called the local action of the pair $(\Gamma, G)$. If this induced group is permutation-isomorphic to a fixed transitive permutation group L — so that the valency of Γ equals the degree of L — the pair $(\Gamma, G)$ is called locally-L. A transitive permutation group L is graph-restrictive if there is a constant c(L) with $|G_x|\le c(L)$ for every locally-L pair $(\Gamma, G)$. Tutte's 1947 theorem that a connected arc-transitive graph of valency 3 has vertex stabiliser of order dividing 48 says exactly that the two transitive groups of degree 3 are graph-restrictive, and such bounds underlie the finite censuses of symmetric graphs. They are not automatic: connected 4-valent arc-transitive graphs can have arbitrarily large vertex stabilisers, so a genuine hypothesis on the local action is necessary.

Weiss conjectured in 1978 that every finite primitive permutation group (a transitive group preserving no nontrivial partition of the point set) is graph-restrictive; the corresponding statement for quasiprimitive groups (transitive groups in which every nontrivial normal subgroup is transitive) was later conjectured by Praeger. The deepest supporting result, due to Trofimov and Weiss, is that 2-transitive groups are graph-restrictive; its proof uses the Classification of Finite Simple Groups. Potočnik, Spiga and Verret proved in 2012 that a graph-restrictive group is necessarily semiprimitive — transitive, and every normal subgroup is either transitive or semiregular (a subgroup is semiregular when only its identity fixes a point) — and proposed the converse, now called the Potočnik–Spiga–Verret (PSV) conjecture: a finite permutation group is graph-restrictive if and only if it is semiprimitive. Since primitive and quasiprimitive groups are semiprimitive, the PSV conjecture subsumes both earlier conjectures.

This problem concerns the PSV conjecture in the class isolated by Giudici and Morgan: semiprimitive groups admitting a regular normal nilpotent subgroup R. A regular subgroup is transitive with trivial point stabilisers, so its order equals the degree; a nilpotent group is a finite direct product of its Sylow subgroups, so this generalises the regular elementary abelian normal subgroup of an affine permutation group. Weiss had proved in 1979, by his p-factorisation method, a structure theorem for vertex stabilisers of locally-affine graphs, where the local action is primitive affine; Giudici and Morgan asked what survives when the local action K is merely semiprimitive with a regular normal nilpotent subgroup R. They proved (Theorem 1.1) that if the valency d, which equals $|R|$, is coprime to 6, then the edge-kernel $G_{xy}^{[1]}$ — the subgroup of the stabiliser of the edge $\{x,y\}$ that fixes both neighbourhoods $\Gamma(x)$ and $\Gamma(y)$ pointwise — is trivial and consequently $|G_x|\le d!(d-1)!$, so K is graph-restrictive. Their structure theorem for the remaining case (Theorem 1.2, Corollary 1.3) shows that if $G_{xy}^{[1]}\ne 1$ for some edge, then $G_{xy}^{[1]}$ is a p-group with $p\in\{2,3\}$, and K must contain normal subgroups $F<R<J$ with $J/F$ a direct product of copies of $\mathrm{Sym}(3)$ when $p=2$, or of copies of $\mathrm{Alt}(4)$ when $p=3$. Thus a counterexample to the PSV conjecture within this class can only lurk in these exceptional configurations, all of which have degree divisible by 2 or 3.

The paper isolates concrete families where the exceptional configurations occur and where graph-restrictiveness was left undecided: (i) Frobenius groups $K=P\rtimes C_2$ with P an abelian 3-group and $C_2$ acting by inversion, in the action on the cosets of $C_2$ (degree $|P|$) — when P is elementary abelian the group is graph-restrictive (Giudici–Morgan 2014), and when P is cyclic it is a dihedral group of odd degree, also graph-restrictive (Potočnik–Spiga–Verret, Example 11, with c(L)=16); the case left open is P non-cyclic with nontrivial Frattini subgroup $\Phi(P)$ (the intersection of the maximal subgroups of P), the smallest such group having $P\cong C_9\times C_3$ and degree 27; (ii) groups $K=W\rtimes H$ with W a direct sum of $m>1$ copies of the vector space $V=\mathbb{F}_4$ or $V=\mathbb{F}_2^2$ and $H=\mathrm{GL}(V)$ acting naturally on each copy, in the action on the vector set W (degree $4^m$); (iii) groups $K=(V_1\times\cdots\times V_r)\rtimes C_3$ with $V_1=Q_8$ (the quaternion group of order 8) and $V_i$ extraspecial of order $p_i^3$ for primes $p_i\equiv -1\pmod 3$ (an extraspecial p-group is a non-abelian p-group whose centre, derived subgroup and Frattini subgroup coincide and have order p), with $C_3$ acting irreducibly on each $V_i/\mathrm{Z}(V_i)$, in the action on the cosets of $C_3$.

In their subsequent structure theory of semiprimitive groups, Giudici and Morgan proved that a semiprimitive group with at least two plinths (minimal transitive normal subgroups) is graph-restrictive, which reduces the PSV conjecture to semiprimitive groups with a unique plinth. The exceptional regular-nilpotent configurations above lie on the unresolved side of this reduction, and the authors describe the PSV conjecture as possibly the most intractable problem of that theory.

Problem Statement

Decide whether every finite semiprimitive permutation group that admits a regular normal nilpotent subgroup R is graph-restrictive, in the terminology of the background: either prove that there is a constant $c(L)$, depending only on the local action L, such that $|G_x|\le c(L)$ for every locally-L pair $(\Gamma, G)$ with L in this class, or exhibit a group L in the class together with an infinite family of locally-L pairs $(\Gamma_n, G_n)$ whose vertex stabiliser orders $|(G_n)_x|$ are unbounded.

The case where the degree $|R|$ is coprime to 6 was settled affirmatively by Giudici–Morgan (Theorem 1.1: then the edge-kernel $G_{xy}^{[1]}$ is trivial and $|G_x|\le d!(d-1)!$), so the task is precisely the residue in which 2 or 3 divides $|R|$. By their Corollary 1.3, any failure in this residue must occur within the exceptional configurations: local actions K containing normal subgroups $F<R<J$ with $J/F$ a direct product of copies of $\mathrm{Sym}(3)$ (when the edge-kernel is a 2-group) or of $\mathrm{Alt}(4)$ (when it is a 3-group). The paper highlights explicit families within these configurations where graph-restrictiveness was left undecided (Examples 1.4–1.7):

1. the Frobenius groups $K=P\rtimes C_2$ with P a non-cyclic abelian 3-group with $\Phi(P)\ne 1$, acting on the cosets of $C_2$ (the cyclic cases are dihedral groups of odd degree and are graph-restrictive by Potočnik–Spiga–Verret's Example 11; smallest open instance: $P\cong C_9\times C_3$, of degree 27);
2. the groups $K=W\rtimes \mathrm{GL}(V)$ with $V=\mathbb{F}_4$ or $V=\mathbb{F}_2^2$ and $W=V^m$ with $m>1$, acting on the vector set W (degrees $4^m\ge 16$);
3. the groups $K=(Q_8\times V_2\times\cdots\times V_r)\rtimes C_3$ with $V_i$ extraspecial of order $p_i^3$ for primes $p_i\equiv -1\pmod 3$, acting on the cosets of $C_3$.

A negative answer consists of one counterexample and would refute the Potočnik–Spiga–Verret conjecture as a whole, since that conjecture predicts that every semiprimitive group, in particular every group in this class, is graph-restrictive. The determination is required for the entire class of semiprimitive groups with a regular normal nilpotent subgroup.

References

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