For a finite group $G$, the Hurwitz space $\mathcal{H}^{in}_{r,g}(G)$ is the space of genus $g$ covers of the Riemann sphere with $r$ branch points and the monodromy group $G$. In this paper, we give a complete list of primitive genus one systems of affine type. That is, we assume that $G$ is a primitive group of affine type. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in $\mathcal{H}^{in}_{r,1}(G)$. Furthermore, we give a new algorithm for computing large braid orbits on Nielsen classes. This algorithm utilizes a correspondence between the components of $\mathcal{H}^{in}_{r,1}(G)$ and $\mathcal{H}^{in}_{r,1}(M)$, where $M$ is the point stabilizer in $G$.
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.
A permutation group is {\it binary} if its orbits on $k$-tuples, for any integer $k\geq 2$, can be deduced from its orbits on $2$-tuples. Cherlin conjectured that a finite primitive binary permutation group $G$ must lie in one of three known families. In this paper we complete the proof of this conjecture. To do this we study the case where the group $G$ is almost simple of Lie type.
Let $G$ be a finite primitive permutation group on a set $Ω$ with nontrivial point stabilizer $G_α$. We say that $G$ is extremely primitive if $G_α$ acts primitively on each of its orbits in $Ω\setminus \{α\}$. In earlier work, Mann, Praeger and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall's conjecture on the number of maximal sub…
Suppose that $G$ is a finite, transitive, solvable permutation group acting on a set $S$ with $n$ elements. Let $G_0$ be the stabilizer of a point $α\in Ω$. Define the rank of a permutation group, denoted $r(G),$ as the number of distinct orbits of $G_0$ in $S$ (including the trivial orbit $\{α\}$). Huppert \cite{Huppert} and Foulser \cite{Foulser} classified all finite, solvable, permutation groups of rank two and three respectively, and Foulser restricted the rank four groups to a small list of possibilities. This paper completes the classification of all groups of rank less than $5$ by expl…
Let $G$ be a finite permutation group on $Ω$. An ordered sequence of elements of $Ω$, $(ω_1,\dots, ω_t)$, is an irredundant base for $G$ if the pointwise stabilizer $G_{(ω_1,\dots, ω_t)}$ is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same size we say that $G$ is an IBIS group. In this paper we show that if a primitive permutation group is IBIS, then it must be almost simple, of affine-type, or of diagonal type. Moreover we prove that a diagonal-type primitive permutation groups is IBIS if and only if it is isomorphic to $PSL(2,…
This paper presents an analysis of primitive permutation groups of degree $3p$, where $p$ is a prime number, analogous to H. Wielandt's treatment of groups of degree $2p$. It is also intended as an example of the systematic use of combinatorial methods as surveyed in \S6 for distilling information about a permutation group from knowledge of the decomposition of its character. The work is organised into three parts. Part I contains the lesser half of the calculation, the determination of the decomposition of the permutation character. Part II contains a survey of the combinatorial methods and, …
Let G be a permutation group, acting on a set Ωof size n. A subset B of Ωis a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there exist integers m and r \geq 1 such that Alt(m)^r \unlhd G \leq Sym(m) \wr Sym(r), where the action of Sym(m) is on k-element subsets of {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 M24 in its natural action on 24 points, or b(G) \leq \lceil \log n\rceil+1. F…
We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}Γ\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.
We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the $2$-homogeneous and the primitive groups. Lastly, we define $k$-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a $k$-reachable permutation group of degree $n$ with $6 \le k \le n - 6$ is eit…
Let $G$ be a finite non-regular primitive permutation group on a set $Ω$ with point stabiliser $G_α$. Then $G$ is said to be extremely primitive if $G_α$ acts primitively on each of its orbits in $Ω\setminus \{α\}$, which is a notion dating back to work of Manning in the 1920s. By a theorem of Mann, Praeger and Seress, it is known that every extremely primitive group is either almost simple or affine, and all the almost simple examples have subsequently been determined. Similarly, Mann et al. have classified all of the affine extremely primitive groups up to a finite, but undetermined, collect…
Let $G$ be a permutation group on a set $Ω$ of size $t$. We say that $Λ\subseteqΩ$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $Λ$. We define the height of $G$ to be the maximum size of an independent set, and we denote this quantity $\mathrm{H}(G)$. In this paper we study $\mathrm{H}(G)$ for the case when $G$ is primitive. Our main result asserts that either $\mathrm{H}(G)< 9\log t$, or else $G$ is in a particular well-studied family (the "primitive large--base groups"). An immediate corollary of this result is a character…
Let $G$ be a permutation group on a set $Ω$. A base for $G$ is a subset of $Ω$ whose pointwise stabiliser is trivial, and the base size of $G$ is the minimal cardinality of a base. If $G$ has base size $2$, then the corresponding Saxl graph $Σ(G)$ has vertex set $Ω$ and two vertices are adjacent if they form a base for $G$. A recent conjecture of Burness and Giudici states that if $G$ is a finite primitive permutation group with base size $2$, then $Σ(G)$ has the property that every two vertices have a common neighbour. We investigate this conjecture when $G$ is an affine group and a point sta…
General bounds are presented for the diameters of orbital graphs of finite affine primitive permutation groups. For example, it is proved that the orbital diameter of a finite affine primitive permutation group with a nontrivial point stabilizer $ H \leq \mathrm{GL}(V) $, where the vector space $ V $ has dimension $ d $ over the prime field, can be bounded in terms of $ d $ and $ \log |V| / \log |H| $ only. Several infinite families of affine primitive permutation groups with large orbital diameter are constructed. The results are independent from the classification of finite simple groups.
Let $G$ be a finite primitive permutation group on a set $Ω$ and recall that the fixed point ratio of an element $x \in G$, denoted ${\rm fpr}(x)$, is the proportion of points in $Ω$ fixed by $x$. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing ${\rm fpr}(x)$ with the order of $x$. Our main theorem classifies the triples $(G,Ω,x)$ as above with the property that $x$ has prime order $r$ and ${\rm fpr}(x) > 1/(r+1)$. There are several applications. Firstly, we extend earlier work of Guralni…
