On the number of conjugacy classes of a primitive permutation group

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$ . We prove that either $k(G)< n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$ , or $G$ belongs to explicit families of examples.

Jordan permutation groups and limits of 𝐷-relations

We construct via Fraïssé amalgamation an 𝜔-categorical structure whose automorphism group is an infinite oligomorphic Jordan primitive permutation group preserving a “limit of 𝐷-relations”. The construction is based on a semilinear order whose elements are labelled by sets carrying a 𝐷-relation, with strong coherence conditions governing how these 𝐷-sets are inter-related.

On valency problems of Saxl graphs

Let 𝐺 be a permutation group on a set Ω, and recall that a base for 𝐺 is a subset of Ω such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of 𝐺, denoted Σ ⁢ ( G ) \Sigma(G) , with vertex set Ω and two vertices adjacent if and only if they form a base for 𝐺. If 𝐺 is transitive, then Σ ⁢ ( G ) \Sigma(G) is vertex-transitive, and it is natural to consider its valency (which we refer to as the valency of 𝐺). In this paper, we present a general method for computing the valency of any finite transitive group, and we use it to calculate the exact valency of every primitive group with stabiliser a Frobenius group with cyclic kernel. As an application, we calculate the valency of every almost simple primitive group with an alternating socle and soluble stabiliser, and we use this to extend results of Burness and Giudici on almost simple primitive groups with prime-power or odd valency.

Abelian permutation groups with graphical representations

In this paper we characterize those automorphism groups of colored graphs and digraphs that are abelian as abstract groups. This is done in terms of basic permutation group properties. Using Schur’s classical terminology, what we provide is characterizations of the classes of 2-closed and $$2^*$$ 2 ∗ -closed abelian permutation groups. This is the first characterization concerning these classes since they were defined.

The number of set-orbits of a solvable permutation group

A permutation group 𝐺 acting on a set Ω induces a permutation action on the power set P ⁢ ( Ω ) \mathscr{P}(\Omega) (the set of all subsets of Ω). Let 𝐺 be a finite permutation group of degree 𝑛, and let s ⁢ ( G ) s(G) denote the number of orbits of 𝐺 on P ⁢ ( Ω ) \mathscr{P}(\Omega) . In this paper, we give the explicit lower bound of log 2 ⁡ s ⁢ ( G ) / log 2 ⁡ | G | \log_{2}s(G)/{\log_{2}\lvert G\rvert} over all solvable groups 𝐺. As applications, we first give an explicit bound of a result of Keller for estimating the number of conjugacy classes, and then we combine it with the McKay conjecture to estimate the number of p ′ p^{\prime} -degree irreducible representations of a solvable group.

Indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level 2 with abelian permutation group

We present a construction of all finite indecomposable involutive solutions of the Yang–Baxter equation of multipermutational level at most 2 with abelian permutation group. As a consequence, we obtain a formula for the number of such solutions with a fixed number of elements. We also describe some properties of the automorphism groups in this case; in particular, we show they are regular abelian groups.

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.