Related papers: Derangements in intransitive groups

Let $G$ be a finite almost simple group of Lie type acting faithfully and primitively on a set $\Omega$. We prove an analogue of the Boston--Shalev conjecture for conjugacy classes: the proportion of conjugacy classes of $G$ consisting of…

Given a group $G$, we write $g^G$ for the conjugacy class of $G$ containing the element $g$. A theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the commutator subgroup…

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…

Each group G of nxn permutation matrices has a corresponding permutation polytope, P(G):=conv(G) in R^{nxn}. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then…

Let $G$ be a finite primitive permutation group and let $\kappa(G)$ be the number of conjugacy classes of derangements in $G$. By a classical theorem of Jordan, $\kappa(G) \geqslant 1$. In this paper we classify the groups $G$ with…

We study Cartesian decompositions of sets that are acted upon intransitively by innately transitive permutation groups. We prove that such groups have at most three orbits on such a decomposition. A consequence of this result is that if $G$…

Given a permutation group $G$, the derangement graph of $G$ is defined with vertex set $G$, where two elements $x$ and $y$ are adjacent if and only if $xy^{-1}$ is a derangement. We establish that, if $G$ is transitive with degree exceeding…

In recent years there has been significant progress in the study of products of subsets of finite groups and of finite simple groups in particular. In this paper we consider which families of finite simple groups $G$ have the property that…

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

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…

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups.…

H.J. Zassenhaus conjectured that any unit of finite order and augmentation $1$ in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of $G$. We prove the…

Recent investigations on the set of commutators between the elements of a finite group having relatively prime orders have prompt us to propose a variant of the Ore conjecture: For every finite non-abelian simple group and for every $g\in…

We investigate the proportion of fixed point free permutations (derangements) in finite transitive permutation groups. This article is the first in a series where we prove a conjecture of Shalev that the proportion of such elements is…

Let G,H be closed permutation groups on an infinite set X, with H a subgroup of G. It is shown that if G and H are orbit-equivalent, that is, have the same orbits on the collection of finite subsets of X, and G is primitive but not…

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has…

We discuss Jordan's theorem on finite subgroups of invertible matrices and give an account of his original proof.

A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits for the induced…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

In this paper we prove several results about the lattice of imprimitivity systems of a permutation group containing a cyclic subgroup with at most two orbits. As an application we generalize the first Ritt theorem about functional…