Base sizes of primitive permutation groups

Let G be a permutation group, acting on a set $$\varOmega $$ Ω of size n. A subset $${\mathcal {B}}$$ B of $$\varOmega $$ Ω is a base for G if the pointwise stabilizer $$G_{({\mathcal {B}})}$$ G ( B ) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of $$\mathrm {Sym}(n)$$ Sym ( n ) is large base if there exist integers m and $$r \ge 1$$ r ≥ 1 such that $${{\,\mathrm{Alt}\,}}(m)^r \unlhd G \le {{\,\mathrm{Sym}\,}}(m)\wr {{\,\mathrm{Sym}\,}}(r)$$ Alt ( m ) r ⊴ G ≤ Sym ( m ) ≀ Sym ( r ) , where the action of $${{\,\mathrm{Sym}\,}}(m)$$ Sym ( m ) is on k-element subsets of $$\{1,\dots ,m\}$$ { 1 , ⋯ , m } and the wreath product acts with product action. In this paper we prove that if G is primitive and not large base, then either G is the Mathieu group $$\mathrm {M}_{24}$$ M 24 in its natural action on 24 points, or $$b(G)\le \lceil \log n\rceil +1$$ b ( G ) ≤ ⌈ log n ⌉ + 1 . Furthermore, we show that there are infinitely many primitive groups G that are not large base for which $$b(G) > \log n + 1$$ b ( G ) > log n + 1 , so our bound is optimal.

ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

INVARIANT MEASURES CONCENTRATED ON COUNTABLE STRUCTURES

Let$L$be a countable language. We say that a countable infinite$L$-structure${\mathcal{M}}$admits an invariant measure when there is a probability measure on the space of$L$-structures with the same underlying set as${\mathcal{M}}$that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of${\mathcal{M}}$. We show that${\mathcal{M}}$admits an invariant measure if and only if it has trivial definable closure, that is, the pointwise stabilizer in$\text{Aut}({\mathcal{M}})$of an arbitrary finite tuple of${\mathcal{M}}$fixes no additional points. When${\mathcal{M}}$is a Fraïssé limit in a relational language, this amounts to requiring that the age of${\mathcal{M}}$have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.

FINITE RIGID SETS IN CURVE COMPLEXES

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex 𝔛 of the curve complex [Formula: see text] such that every locally injective simplicial map [Formula: see text] is the restriction of an element of [Formula: see text], unique up to the (finite) pointwise stabilizer of 𝔛 in [Formula: see text]. Furthermore, if S is not a twice-punctured torus, then we can replace [Formula: see text] in this statement with the extended mapping class group Mod ±(S).

The Cost of 2-Distinguishing Cartesian Powers

A graph $G$ is said to be $2$-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the label classes. The minimum size of a label class in any such labeling of $G$ is called the cost of $2$-distinguishing $G$ and is denoted by $\rho(G)$. The determining number of a graph $G$, denoted $\det(G)$, is the minimum size of a set of vertices whose pointwise stabilizer is trivial. The main result of this paper is that if $G^k$ is a $2$-distinguishable Cartesian power of a prime, connected graph $G$ on at least three vertices with $\det(G)\leq k$ and $\max\{2, \det(G)\} < \det(G^k)$, then $\rho(G^k) \in \{\det(G^k), \det(G^k)+1\}$. In particular, for $n\geq 3$, $\rho(K_3^n)\in \{ \left\lceil {\log_3 (2n+1)} \right\rceil$ $+1, \left\lceil {\log_3 (2n+1)} \right\rceil$ $+2\}$.

Invariants of reflection groups, arrangements, and normality of decomposition classes in Lie algebras

Suppose that W is a finite, unitary, reflection group acting on the complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of W-invariant polynomial functions on V to the algebra of C-invariant functions on X. In this note we consider the special case when W is a Coxeter group, V is the complexified reflection representation of W, and X is in the lattice of the arrangement of W, and give a simple, combinatorial characterization of when the restriction mapping is surjective in terms of the exponents of W and C. As an application of our result, in the case when W is the Weyl group of a semisimple, complex Lie algebra, we complete a calculation begun by Richardson in 1987 and obtain a simple combinatorial characterization of regular decomposition classes whose closure is a normal variety.

Graphs with a locally linear group of automorphisms

Let Γ be an undirected graph, V(Γ) the vertex set of Γ and G a subgroup of aut(Γ). For each vertex x ↦ V(Γ), let Γx denote the set of vertices adjacent to x in Γ and the permutation group induced on Γx. by the stabilizer Gx. For each i ≥ 1, will denote the pointwise stabilizer in Gx of the set of vertices at distance at most i from x in Γ. Letfor each i ≥ 1 and any set of vertices x, y, …, z of Γ. An s-path (or s-arc) is an (s + 1)-tuple (x0, x1, … xs) of vertices such that xi ↦ Γxi–1 for 1 ≤ i ≤ s and xi ╪ xi–2 for 2 ≤ i ≤ s.

Subgroups of small index in infinite symmetric groups. II

Throughout this paper κ denotes an infinite cardinal, S = Sym(κ) and G is a subgroup of S. We shall be seeking the subgroups G with [S: G] < 2κ. In [2], the following result was proved.Theorem 1. If [S: G] ≤ κthen there exists a subset Δ of k such that ∣Δ∣ < k and S(Δ) ≤ G.Here S(Δ) = Sym(K/⊿) is the pointwise stabilizer of Δ in S.However, the converse of Theorem 1 is not true. For if cf(κ) ≤ ∣Δ∣ < κ, then [S: S(Δ)] ≥ κcf(κ) > κ. This suggests that a substantially sharpened version of Theorem 1 may be true.Question 1 [2]. Is it provable in ZFC, or even in ZFC with GCH, that if [S: G] ≤ κ then there is a subset Δ of κ such that ∣Δ∣ < cf(κ) and S(Δ) ≤ G?At least two of the authors of [2] made a serious attempt to answer the above question positively. In §3, we shall see that they were essentially trying to prove that measurable cardinals do not exist.The following result, due independently to Semmes [5] and Neumann [2], suggests a second way in which Theorem 1 might be improved.Theorem 2. If k = ℵ0andthen there is a finite subset Δ of k such thatS(Δ) ≤ G.Question 2 [2]. Is it provable in ZFC that if [S: G] < 2κ then there is a subset Δ of κ such that ∣Δ∣ < κ and S(Δ) < G?This question will be answered negatively in §4.