for an−1 · · · a1a0 ∈ V (Hk
n ). Denote by G(Sk) the set {gσ | σ ∈ Sk}. G(Sk) is canonically
isomorphic to Sk . In step 1 that follows, we show that any element of G(Sk) is an
element of G(Hk
n ). We then show, in step 2, that each element of G(Hk
n ) arises as an
element of G(Sk), completing the proof of the main theorem.