The strong Gelfand pairs of the Suzuki family of groups

TL;DR

This work classifies strong Gelfand subgroups within the Suzuki family Sz$(q)$. By leveraging the reduction that eliminates non-strong pairs through inclusion of maximal subgroups, the authors analyze the known maximal subgroups using character-degree arguments and a total-character criterion. They show that for all $q>2$, Sz$(q)$ has no nontrivial strong Gelfand subgroups, with explicit non-strongness results for each maximal candidate, while Sz$(2)$ admits exactly four strong Gelfand subgroups: Sz$(2)$, $D_{10}$, $C_5$, and $C_4$. The results connect the representation-theoretic structure of Sz$(q)$ to its subgroup lattice and provide a complete profile of strong Gelfand phenomena within this infinite family of groups.

Abstract

We find every subgroup $H\leq Sz(q)$ so that the pair $(Sz(q), H)$ is a strong Gelfand pair.

Theorems & Definitions (16)

• Definition 1.1
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 3.1
• proof
• Lemma 3.2
• proof