§ Schrier sims --- why purify generators times coset

• Let p = (0 3 4)(1 2). Let G =

  . What is the stabilizer of k=0?

• purify(p) = e so we would imagine we would have H = e.
• But actually, consider orbit(k). We have 0 <-> id, 3 <-> p, 4 <-> p^2.
• If I now consider p * orbit(k) then I get p, p^2, p^3, where purify(p) = id, purify(p^2) = id, purify(p^3) = p^3.
• Thus we find the nontrivial generator p^3.