11 problems

Let $G$ be a primitive permutation group, and let $d$ be a positive integer. Consider the nondiagonal orbital digraphs of $G$, and suppose that their minimal out-valency is at most…

DMS's conjecture. There exists an interval $X$ of positive integers, not containing $1$, such that no primitive group $G$ acting on any domain $\Omega$ satisfies

Lee–Spiga conjecture. Every almost simple primitive IBIS group is one of the groups in Table 1 of Lee and Spiga.

Let $G$ range over the groups listed in Table 2 of the source, namely the remaining socles of primitive groups of degree $pq$ that do not admit imprimitive subgroups, where $p$ and…

Generalized Sims conjecture. There exists a function $g:\mathbb{N}\to\mathbb{N}$ such that, if $G$ is a finite primitive group with a suborbit $\beta^{G_\alpha}$ of length $d>1$, t…

Let $G\leqslant \operatorname{Sym}(\Omega)$ be a primitive permutation group. The problem asks whether there exist $\alpha,\beta\in\Omega$ such that … Here $G_\alpha$ is the stabil…

Primitive-subgroup conjecture. Almost surely, $\langle\pi,\pi'\rangle$ is not contained in any primitive subgroup apart from $S_n$, $A_n$, and any conjugate of…

Let $G$ be a primitive permutation group acting on a set $\Omega$ of cardinality $n\ge 8$. For a subset $S\subseteq \Omega$, write $S^G$ for its orbit under $G$. Siemons–Wagner con…

Regular-cycle proportion conjecture. There exists an absolute positive constant $c$ such that, for every such $(G,\Omega)$ and every $g\in G$,

Giudici–Praeger–Spiga conjecture. If some element of $(G,\Omega)$ has no regular cycle, then there exist integers $k\geq1$, $r\geq1$, and $m\geq5$ such that $G$ preserves a product…

Let $S_n$ be the symmetric group, and consider the proportion of elements of $S_n$ that belong to a primitive subgroup not containing $A_n$. The paper proves an upper bound of…