On Wiegold's conjecture for the small Ree groups

TL;DR

The paper proves that the small Ree groups $^2\mathsf{G}_2(3^{2e+1})$ satisfy the Wiegold conjecture for $k$-tuples with $k\ge 5$, by exploiting the natural action on the Ree unital $U_R(q)$ and a detailed elimination procedure for structural subgroups. The strategy proceeds in stages: (i) reduce generating triples to non-involutive, non-unipotent generators, (ii) eliminate all structural subgroups by moving generators away from point- and block-stabilizers via Nielsen moves, and (iii) inductively connect to redundant vectors, culminating in connectivity of the extended PRA graph on generating tuples. A central role is played by the rank-2 geometry of $U_R(q)$, explicit $7\times7$ matrix realizations of point stabilizers from the model in TTM07, and a careful case analysis of centralizers and subgroup inclusion in $^2\mathsf{G}_2(q)$. The result extends the Wiegold framework to rank-one finite simple groups of Lie type, illustrating a robust method for handling complex subgroup lattices in exceptional groups and contributing to the understanding of Aut$(F_n)$ actions on finite simple groups.

Abstract

The Wiegold conjecture holds for the small Ree groups for $k$-tuples where $k \geq 5$.

Theorems & Definitions (48)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof