Related papers: Intransitive Cartesian decompositions preserved by…

Covariant or invariant functions under a compact linear group can be expressed in terms of functions defined in the orbit space of the group. The semialgebraic relations defining the orbit spaces of all finite coregular real linear groups…

Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$, and $G$ is not metacyclic. Then $G$ always has a regular orbit on $V$ except for a few "small" cases. We completely…

Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut(G) on the set of maximal independent subsets of G determine the indecomposable decompositions of G. G contains a direct sum of pure strongly…

Let $H \leq S_n$ be an intransitive group with orbits $\Omega_1, \Omega_2, \ldots ,\Omega_k$. Then certainly $H$ is a subdirect product of the direct product of its projections on each orbit, $H|_{\Omega_1} \times H|_{\Omega_2} \times…

A classical theorem of Jordan asserts that if a group $G$ acts transitively on a finite set of size at least $2$, then $G$ contains a derangement (a fixed-point free element). Generalisations of Jordan's theorem have been studied…

We study the orbits under the natural action of a permutation group $G \subseteq S_n$ on the powerset $\mathscr{P}(\{1, \dots , n\})$. The permutation groups having exactly $n+1$ orbits on the powerset can be characterized as set-transitive…

We study $(G,2)$-arc-transitive graphs for innately transitive permutation groups $G$ such that $G$ can be embedded into a wreath product $\sym\Gamma\wr\sy\ell$ acting in product action on $\Gamma^\ell$. We find two such connected graphs:…

The finite orbits of the braid group action on Stokes matrices are studied and are shown to be the orbits on ordered sets of reflections, generating finite groups. All invariants of a reflection arrangement are determined. Determination of…

Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $...\wr G_2\wr G_1$ is topologically finitely generated if…

Let $G$ be a nontrivial transitive permutation group on a finite set $\Omega$. An element of $G$ is said to be a derangement if it has no fixed points on $\Omega$. From the orbit counting lemma, it follows that $G$ contains a derangement,…

Let $G$ be a permutation group on a finite set $\Omega$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ having the same orbits as $G$ on the $k$-th Cartesian power $\Omega^k$ of $\Omega$.…

An $H$-decomposition of a graph $\Gamma$ is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of $H$-decomposition that is highly symmetrical in the sense that the subgraphs (copies…

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the…

Let $G$ be a Lie group and let $M$ be a proper smooth $G$-manifold. If $M$ is connected and $\dim(M)\geq 2$, the group of diffeomorphisms of $M$, that are isotopic to the identity through a compactly supported isotopy, acts $n$-transitively…

We study finite transitive permutation groups $G\leqslant\operatorname{Sym}(\Omega)$ such that all orbits of the conjugation action on $G$ of the normaliser of $G$ in $\operatorname{Sym}(\Omega)$ have size bounded by some constant. Our…

Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive…

Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in $\mathcal{S}$ is…

We say that a group G acts infinitely transitively on a set X if for every integer m the induced diagonal action of G is transitive on the cartesian mth power of X with the diagonals removed. We describe three classes of affine algebraic…

A group is said to be self-similar provided it admits a faithful state-closed representation on some regular $m$-tree and the group is said to be transitive self-similar provided additionally it induces transitive action on the first level…

The transitivity degree of a group $G$ is the supremum of all integers $k$ such that $G$ admits a faithful $k$-transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The…