相关论文: Irredundant bases for the symmetric group

A base for a permutation group $G$ acting on a set $\Omega$ is a sequence $\mathcal{B}$ of points of $\Omega$ such that the pointwise stabiliser $G_{\mathcal{B}}$ is trivial. The base size of $G$ is the size of a smallest base for $G$.…

A base of a permutation group (X,G) is a subset B of X such that its pointwise stabilizer is the trivial group. A list (x1,x2, ... ,xk) of elements of X is irredundant if each element is not in the pointwise stabilizer of its predecessors.…

Fix a positive integer $d$ and let $\Gamma_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $\Gamma_d$ were studied by Babai, Cameron and P\'{a}lfy in the 1980s and they determined…

Let $G$ be a finite permutation group on $\Omega.$ An ordered sequence $(\omega_1,\dots, \omega_t)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its…

We show that the minimal base size $b(G)$ of a finite primitive permutation group $G$ of degree $n$ is at most $2 (\log |G|/\log n) + 24$. This bound is asymptotically best possible since there exists a sequence of primitive permutation…

Let $G$ be a permutation group on a finite set $\Omega$. The base size of $G$ is the minimal size of a subset of $\Omega$ with trivial pointwise stabiliser in $G$. In this paper, we extend earlier work of Fawcett by determining the precise…

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 base for a subgroup $G$ of $\mathrm{Sym}(\Omega)$ is a sequence of elements of $\Omega$ with trivial pointwise stabiliser. The size of the smallest base for $G$ is denoted $b(G)$. There is a natural greedy algorithm to compute a base for…

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…

Let S=Sym(\Omega) be the group of all permutations of a countably infinite set \Omega, and for subgroups G_1, G_2\leq S let us write G_1\approx G_2 if there exists a finite set U\subseteq S such that < G_1\cup U > = < G_2\cup U >. It is…

Let $G$ be a permutation group on a finite set $\Omega$. A subset $B \subseteq \Omega$ is a base for $G$ if the pointwise stabilizer of $B$ in $G$ is trivial. The base size of $G$, denoted $b(G)$, is the smallest size of a base. A well…

Let $G$ be a transitive permutation group on a finite set $\Omega$ and recall that a base for $G$ is a subset of $\Omega$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$…

Let $G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base…

Let $G$ be a transitive permutation group on a finite set with solvable point stabiliser and assume that the solvable radical of $G$ is trivial. In 2010, Vdovin conjectured that the base size of $G$ is at most 5. Burness proved this…

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$…

Let $G$ be a finite group and recall that the Frattini subgroup ${\rm Frat}(G)$ is the intersection of all the maximal subgroups of $G$. In this paper, we investigate the intersection number of $G$, denoted $\alpha(G)$, which is the minimal…

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…

A group $\Gamma$ is said to be uniformly HS stable if any map $\varphi : \Gamma \to U(n)$ that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We…

We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i),…

Let $G$ be a permutation group on the finite set $\Omega$. We prove various results about partitions of $\Omega$ whose stabilizers have good properties. In particular, in every solvable permutation group there is a set-stabilizer whose…