Related papers: Erd\H{o}s-Ko-Rado problems for permutation groups

The study of intersecting structures is central to extremal combinatorics. A family of permutations $\mathcal{F} \subset S_n$ is \emph{$t$-intersecting} if any two permutations in $\mathcal{F}$ agree on some $t$ indices, and is…

We conjecture that if $G$ is a finite primitive group and if $g$ is an element of $G$, then either the element $g$ has a cycle of length equal to its order, or for some $r,m$ and $k$, the group $G\leq S_m\wr S_r$, preserving a product…

A transitive permutation group of prime degree is doubly transitive or solvable. We give a direct proof of this theorem by Burnside which uses neither S-ring type arguments, nor representation theory.

In this paper we prove an Erd\H{o}s-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL(3,q), in its natural action on the points of the…

Let $G$ be a permutation group on a set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted…

Let $G$ be a transitive permutation group on $\Omega$ containing two points $\alpha, \beta$ such that $G_{\alpha}\cap G_{\beta}=1$. The Saxl graph $\Sigma(G)$ of $(G, \Omega)$ is defined as the graph with vertex set $\Omega$, where two…

Let $G$ be a permutation group acting on a finite set $\Omega$ of cardinality $n$. The number of orbits of the induced action of $G$ on the set $\Omega_m$ of all size $m$ subsets of $\Omega$ satisfies the trivial inequalities…

We treat the problem of finding transitive subgroups G of S_n containing normal subgroups N_1 and N_2, with N_1 transitive and N_2 not transitive, such that G/N_1 is isomorphic G/N_2. We show that such G exist whenever n has a prime factor…

The minimal degree of a permutation group $G$ is defined as the minimal number of non-fixed points of a non-trivial element of $G$. In this paper we show that if $G$ is a transitive permutation group of degree $n$ having no non-trivial…

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer in $G$ is trivial. By $b(G)$ we denote the size of the smallest base of $G$. Every permutation group with $b(G)=2$ contains…

We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group,…

Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on $\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that $G$ \emph{synchronizes} $f$ if the transformation semigroup $\langle G,f\rangle$ contains a…

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts…

Let $G$ be a transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G|…

Let $G$ be a closed permutation group on a countably infinite set $\Omega$, which acts transitively but not highly transitively. If $G$ is oligomorphic, has no algebraicity and weakly eliminates imaginaries, we prove that any probability…

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite permutation group and recall that the base size of $G$ is the minimal size of a subset of $\Omega$ with trivial pointwise stabiliser. There is an extensive literature on base sizes for…

This paper classifies quasiprimitive permutation groups with a transitive subgroup which is isomorphic to $\A_n$ for some $n\geqslant5$.

This article began as a study of the structure of infinite permutation groups G in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point…

A group G is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is…

The first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian…