The $(2,3)$-generation of the finite simple odd-dimensional orthogonal groups

TL;DR

This work addresses the problem of $(2,3)$-generation for finite simple groups by providing explicit $(2,3)$-generators for odd-dimensional orthogonal groups $\Omega_{2k+1}(q)$ with $k\ge 4$, and extending to several even-dimensional families under congruence constraints. The authors build generators of orders $2$ and $3$ with a parameter $a\in \mathbb{F}_q^*$, prove irreducibility, and exclude containment in maximal subgroups using trace and eigenvalue arguments, as well as subspace-invariant techniques. Key contributions include constructive generation for $n\in\{9,11,13,17\}$ and broad coverage for $n\ge 12$ with explicit matrix families, culminating in corollaries that cover many even-dimensional cases; the paper also delineates remaining open orthogonal cases. The results have practical impact for the explicit realization and testing of generation in computational group theory, advancing the classification of $(2,3)$-generated finite simple groups.

Abstract

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups $Ω_{2k+1}(q)$, $k\geq 4$. As a byproduct we also obtain $(2,3)$-generators for $Ω_{4k}^+(q)$ with $k\geq 3$ and $q$ odd, and for $Ω_{4k+2}^\pm(q)$ with $k\geq 4$ and $q\equiv \pm 1 \pmod 4$.

Figures (2)

• Figure 1: Generators of $\Omega_9(q)$.
• Figure 2: Alternative generators of $\Omega_9(q)$.

Theorems & Definitions (45)

• Theorem 1.1
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3