Related papers: Imprimitive Permutations in Primitive Groups

In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime…

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.

The goal of invariant theory is to find all the generators for the algebra of representations of a group that leave the group invariant. Such generators will be called \emph{basic invariants}. In particular, we set out to find the set of…

Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner…

We propose a quantitative notion of permutation stability for finitely generated groups. Our notion is related to, but distinct from, the ``stability rate'' introduced by Becker and Mosheiff (which is valid within the class of finitely…

We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also classify finite almost simple groups that have…

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.

We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of…

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…

This is an introduction to the finite groups, with focus on the groups of permutations and reflections, and more generally, on the finite groups of unitary matrices. We first discuss the basics of group theory, featuring the cyclic,…

Working in a theory with an integer-valued dimension on interpretable sets, we classify pseudofinite definably primitive permutation groups acting on one-dimensional sets which satisfy a version of chain condition on centralizers and on…

Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources of (primitive) commutative association schemes, * collect…

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups…

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$ and recall that an element of $G$ is a derangement if it has no fixed points. Derangements always exist by a classical theorem of Jordan, but there are so-called…

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 $\Omega \setminus \{\alpha\}$.…

We prove that an infinite family of semiprimitive groups are graph-restrictive. This adds to the evidence for the validity of the PSV Conjecture and increases the minimal imprimitive degree for which this conjecture is open to 12. Our…

We present the ideas behind an algorithm to compute normalizers of primitive groups with non-regular socle in polynomial time. We highlight a concept we developed called permutation morphisms and present timings for a partial implementation…

The computation of the normaliser of a permutation group in the full symmetric group is an important and hard problem in computational group theory. This article reports on an algorithm that builds a descending chain of overgroups to…

Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from [DEP00] (which correspond to the case where H is…