Generating Finite Coxeter Groups with Elements of the Same Order

Abstract

Supposing $G$ is a group and $k$ a natural number, $d_k(G)$ is defined to be the minimal number of elements of $G$ of order $k$ which generate $G$ (setting $d_k(G) = 0$ if $G$ has no such generating sets). This paper investigates $d_k(G)$ when $G$ is a finite Coxeter group either of type $B_n$ or $D_n$ or of exceptional type. Together with Garzoni [3] and Yu [10], this determines $d_k(G)$ for all finite irreducible Coxeter groups $G$ when 2$ \leq k \leq$rank$(G)$ (rank$(G) + 1$ when $G$ is of type $A_n$).