Further Rigid Triples of Classes in $G_{2}$

TL;DR

This paper proves the existence of specific rigid triples in the algebraic group G2 over characteristic 5, showing that certain finite groups G2(5^n) are not generated by elements of orders 2, 4, and 5, confirming a conjecture.

Contribution

It establishes new rigid triples in G2 in characteristic 5 and confirms a conjecture about the generation properties of G2(5^n).

Findings

Existence of two rigid triples in G2 in characteristic 5.

G2(5^n) groups are not (2,4,5)-generated.

Confirms Marion's conjecture for these groups.

Abstract

We establish the existence of two rigid triples of conjugacy classes in the algebraic group $G_{2}$ in characteristic $5$, complementing results of the second author with Liebeck and Marion. As a corollary, the finite groups $G_{2}(5^n)$ are not $(2,4,5)$-generated, confirming a conjecture of Marion in this case.