In: Advanced Math
(The “conjugation rewrite lemma”.) Let σ and τ be permutations.
(a) Show that if σ maps x to y then στ maps τ(x) to τ(y).
(b) Suppose that σ is a product of disjoint cycles. Show that στ has the same cycle structure as
σ; indeed, wherever (... x y ...) occurs in σ, (... τ(x) τ(y) ...) occurs in στ.
(a) Given that , we have
(b) Suppose that be a cycle in . Thus, we have,
By part a) we have
which shows that is a cycle in . Thus, corresponding to every cycle in we get a cycle in , and conversely, for every cycle in we get a cycle in . Hence, if is a product of cycle, so is , and both has same cycle structure.