Related papers: Base sizes of primitive groups: bounds with explic…

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…

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…

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…

Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $\log |G| / \log n \leq b(G) < 45…

Let $G$ be a permutation group, and denote with $\mu(G)$ and $b(G)$ its minimal degree and base size respectively. We show that for every $\varepsilon>0$, there exists a transitive permutation group $G$ of degree $n$ with \[ \mu(G)b(G) \geq…

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…

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…

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 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 $G \leqslant {\rm Sym}(\Omega)$ 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 transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G|…

In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size, minimal degree, fixity, and chief length of an arbitrary…

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 $G$ 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…

The minimal base size $b(G)$ for a permutation group $G$, is a widely studied topic in the permutation group theory. Z. Halasi and K. Podoski proved that $b(G)\leq 2$ for coprime linear groups. Motivated by this result and the probabilistic…

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…

The minimal degree of a permutation group $G$ is defined as the minimal number of non-fixed points of a non-trivial element of $G$. In this paper we show that if $G$ is a transitive permutation group of degree $n$ having no non-trivial…

We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every…

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 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…