(The “conjugation rewrite lemma”.) Let σ and τ be permutations. (a) Show that if σ maps...

(The “conjugation rewrite lemma”.) Let σ and τ be permutations.

(a) Show that if σ maps x to y then σ^τ maps τ(x) to τ(y).
(b) Suppose that σ is a product of disjoint cycles. Show that σ^τ has the same cycle structure as

σ; indeed, wherever (... x y ...) occurs in σ, (... τ(x) τ(y) ...) occurs in σ^τ.

Solutions

Expert Solution

(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.

Colby Messinger answered 1 year ago

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