6 problems

Let $G$ be a finite primitive non-regular permutation group of degree $n$. Write $\operatorname{fix}(g)$ for the number of points fixed by $g\in G$. Neumann's conjecture. There exi…

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

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 $G$ 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 $g\in G$,