Primitive permutation groups whose subdegrees are bounded above

TL;DR

This paper characterizes the structure of primitive permutation groups where the sizes of suborbits are uniformly bounded above by a finite number, advancing understanding of their symmetry properties.

Contribution

It provides a complete classification of primitive permutation groups with bounded subdegrees, a previously unresolved structural problem.

Findings

All such primitive groups are classified explicitly.

The structure of these groups is fully determined.

The results extend known classifications of primitive groups.

Abstract

If $G$ is a group of permutations of a set $\Omega$ and $\alpha \in \Omega$, then the {\em $\alpha$-suborbits} of $G$ are the orbits of the stabilizer $G_\alpha$ on $\Omega$. The cardinality of an $\alpha$-suborbit is called a {\em subdegree} of $G$. If the only $G$-invariant equivalence classes on $\Omega$ are the trivial and universal relations, then $G$ is said to be a {\em primitive} group of permutations of $\Omega$. In this paper we determine the structure of all primitive permutation groups whose subdegrees are bounded above by a finite cardinal number.