相关论文: Irredundant bases for the symmetric group

Let $\Delta$ be a finite set and $G$ be a subgroup of $\operatorname{Sym}(\Delta)$. An irredundant base for $G$ is a sequence of points of $\Delta$ yielding a strictly descending chain of pointwise stabilisers, terminating with the trivial…

Let $G$ be a finite permutation group acting on a set $\Omega$. An ordered sequence $(\omega_1,\ldots,\omega_\ell)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer of the sequence is trivial and no point…

We prove that the maximum length of an irredundant base for a primitive action of a finite simple group of Lie type is bounded above by a function which is a polynomial in the rank of the group. We give examples to show that this type of…

A base for a permutation group $G$ acting on a set $\Omega$ is a subset $\mathcal{B}$ of $\Omega$ such that the pointwise stabiliser $G_{(\mathcal{B})}$ is trivial. Let $n$ and $r$ be positive integers with $n>2r$. The symmetric and…

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. Denote the minimum size of a base for $G$ by $b(G)$. There is a…

Let G be a permutation group, acting on a set \Omega of size n. A subset B of \Omega is a base for G if the pointwise stabilizer G_(B) is trivial. Let b(G) be the minimal size of a base for G. A subgroup G of Sym(n) is large base if there…

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite almost simple primitive permutation group, with socle $G_0$ and point stabilizer $H$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$,…

Let $G$ be a finite permutation group on $\Omega$. An ordered sequence $(\omega_1\ldots,\omega_\ell)$ 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…

A base B for a finite permutation group G acting on a set X is a subset of X with the property that only the identity of G can fix every point of B. We prove that a primitive diagonal group G has a base of size 2 unless the top group of G…

We study the natural action of $S_n$ on the set of $k$-subsets of the set $\{1,\dots, n\}$ when $1\leq k \leq \frac{n}{2}$. For this action we calculate the maximum size of a minimal base, the height and the maximum length of an irredundant…

In this paper we show that if $G$ is a primitive subgroup of $S_{n}$ that is not large base, then any irredundant base for $G$ has size at most $5 \log n$. This is the first logarithmic bound on the size of an irredundant base for such…

A base for a permutation group $G$ acting on a set $\Omega$ is a subset $\mathcal{B}$ of $\Omega$ whose pointwise stabiliser $G_{(\mathcal{B})}$ is trivial. There is a natural greedy algorithm for constructing a base of relatively small…

Let $G$ be a finite primitive permutation group on a set $\Omega$ with point stabiliser $H$. Recall that a subset of $\Omega$ is a base for $G$ if its pointwise stabiliser is trivial. We define the base size of $G$, denoted $b(G,H)$, to be…

Let $G$ be a permutation group on a set $\Omega$ of size $t$. We say that $\Lambda\subseteq\Omega$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda$. We define the…

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 $G$. We…

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is the minimal cardinality of a base. In this paper we initiate the study of bases for…

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

Let $G$ be a finite permutation group acting on $\Omega$. A base for $G$ is a subset $B \subseteq \Omega$ such that the pointwise stabilizer $G_{(B)}$ is the identity. The base size of $G$, denoted by $b(G)$, is the cardinality of the…

Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if…

An irredundant base $B$ for a permutation group $G\leq \mathrm{Sym}(\Omega)$ is an ordered subset of $\Omega$ with trivial stabiliser such that no base point is fixed by the stabiliser of its predecessors. Groups whose irredundant bases all…