Related papers: Fixed point ratios for finite primitive groups and…

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group and recall that an element in $G$ is a derangement if it has no fixed points on $\Omega$. Let $\Delta(G)$ be the set of derangements in $G$ and define $\delta(G) =…

For a set $\Omega$ an unordered relation on $\Omega$ is a family R of subsets of $\Omega.$ If R is such a relation we let G(R) be the group of all permutations on $\Omega$ that preserves R, that is g belongs to G(R) if and only if x in R…

In this paper we propose a refinement of Sims conjecture concerning the cardinality of the point stabilizers in finite primitive groups and we make some progress towards this refinement. In this process, when dealing with primitive groups…

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm L}_n(q)$ is prime. We present…

A transitive permutation group $G$ on a finite set $\Omega$ is said to be pre-primitive if every $G$-invariant partition of $\Omega$ is the orbit partition of a subgroup of $G$. It follows that pre-primitivity and quasiprimitivity are…

A quasi-semiregular element in a permutation group is an element that has a unique fixed point and acts semiregularly on the remaining points. Such elements were first studied in the context of automorphisms of graphs and occur naturally in…

If G is a finite group, then Pr(G) denotes the fraction of ordered pairs of elements of G which commute. We show that, if l \in (2/9,1] is a limit point of the function Pr on finite groups, then l \in \Q and there exists an e = e_l > 0 such…

Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$.…

In this paper, we study intersecting sets in primitive and quasiprimitive permutation groups. Let $G \leqslant \mathrm{Sym}(\Omega)$ be a transitive permutation group, and ${S}$ an intersecting set. Previous results show that if $G$ is…

We classify all infinite primitive permutation groups possessing a finite point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to all primitive permutation groups with finite point stabilizers.

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many…

Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose $G$ is a finite primitive permutation group on $\Omega$, and $\alpha$, $\beta$ are distinct points of $\Omega$. Does there exist an…

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper we study the involution fixity of almost simple…

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of…

A primary covering of a finite group $G$ is a family of proper subgroups of $G$ whose union contains the set of elements of $G$ having order a prime power. We denote with $\sigma_0(G)$ the smallest size of a primary covering of $G$, and…

A minimal permutation representation of a finite group G is a faithful G-set with the smallest possible size. We study the structure of such representations and show that for certain groups they may be obtained by a greedy construction. In…

We prove that super strongly fractal groups acting on regular rooted trees have null fixed-point proportion. In particular, we show that the fixed-point proportion of an infinite family of iterated monodromy groups of exceptional complex…

We study definably primitive pseudo-finite permutation groups of finite $SU$-rank. We show that if $(G,X)$ is such a permutation group, then the rank of $G$ can be bounded in terms of the rank of $X$, providing an analogue of a theorem of…

In this note we describe a method for finding prime numbers as fixed points of particularly simple sequences. Some basic calculations show that success rates for identifying primes this way are over 99.9%. In particular, it seems that the…

The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. Babai conjectured that if a primitive…