11 problems

Let $G$ be a finite almost simple group with socle $G_0$, and let a non-standard subgroup mean a subgroup whose associated primitive action is non-standard in the sense defined in…

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite permutation group. A base for $G$ is a sequence of points of $\Omega$ with trivial pointwise stabiliser, and $b(G)$ denotes t…

Burness–Giudici conjecture. Every two vertices of $\Sigma(G)$ have a common neighbour.

Let $G\leq \operatorname{Sym}(\Omega)$ be a finite permutation group with base size $b(G)\geq 2$, and let $\mathcal{H}(G)$ be its Saxl hypergraph; in the original graph formulation…

Let $G\leq \operatorname{Sym}(\Omega)$ be a primitive permutation group with base size $b(G)=2$, and let $\Sigma(G)$ be its Saxl graph, whose vertices are the points of $\Omega$ an…

Let $n\geq 2k$, let $G={\rm S}_{n,k}$, and let $H$ be a point stabiliser of $G$. Write $b(G)$ for the base size of the permutation group $G$, let $\mathcal{K}(G,H)$ be the Külshamm…

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite primitive permutation group with base size $b(G)\geqslant 2$. For $\alpha\in\Omega$, call a $G_\alpha$-orbit on $\Omega$ almo…

Let $G$ be a finite group with a core-free maximal subgroup, and let $m(G)$ be the minimal index of a core-free maximal subgroup. Let $R_{\rm \max}(G)$, $R_{\rm ns}(G)$ and…

Logarithmic irredundant-base conjecture. There exists a constant $C>0$ such that, if $G$ is not a primitive large-base group, then

Let $G$ be a finite primitive permutation group with base size $b(G)=2$, and let $\alpha0\Sigma(G)$ be its Saxl graph, whose vertices are the points of the permutation domain and w…

Let $G$ be a finite primitive permutation group on a finite set $S$. A base is a subset $T\subseteq S$ satisfying $G_T=\{e\}$, and $b(G)$ denotes the minimum cardinality of a base.…