On the orbital diameter of classical groups in standard action

TL;DR

The paper investigates the orbital diameter $\mathrm{diam}(X,G)$ for finite primitive groups with standard $t$-actions, focusing on classical groups in standard actions. It proves a general lower bound $\mathrm{diam}(X,G)\ge k$ with $k=\min\{t, n-t\}$ (except in one exceptional orthogonal case where the diameter is $\lfloor k/2\rfloor$) and provides a partial classification of pairs with $\mathrm{diam}(X,G)\le 2$, including a precise equality $\mathrm{diam}(X,G)=k$ for $G_0=\mathrm{PSL}_n(q)$ in case (b) and detailed 2-step results for unitary and orthogonal subcases. The work combines subspace geometry, Witt's lemma, and explicit orbital-graph analyses to bound diameters and identify exact configurations yielding small diameters, advancing understanding of how group action geometry controls orbital connectivity in classical groups.

Abstract

Let $G$ be a primitive permutation group acting on a finite set $X$. The orbital diameter $\mathrm{diam}(X,G)$ is defined to be the supremum of the diameters of the (connected) orbital graphs of $G$ after disregarding the directions of all edges in the graphs. This invariant is studied in the case when $G$ is an almost simple group in a standard action. A lower bound is given for $\mathrm{diam}(X,G)$ and we provide a partial classification of pairs $(X,G)$ for which the orbital diameter is at most $2$.

Theorems & Definitions (32)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 2.1: Witt's lemma
• Lemma 3.1
• proof
• Lemma 3.2
• proof