Let ? be any cycle of length r, and
Let Sn by the set of all bijections Xn -> Xn.
I need help proving two things:
1> that if ? is in Sn, show that ???^-1 is also a cycle of length r.
So far, I know that if ? = (a1 a2 … ar), then I should show that ???^-1 = (?a1 ?a2 … ?ar)
2> if ? is in Sn, show that ? and ???^-1 have the same cycle structure.
(All I know when they say “cycle structure”, is they mean that in the factorization into disjoint cycles have the same number of cycles of each length. Do I need to know something else, too?)
Thanks for any help again.
Let Sn by the set of all bijections Xn -> Xn.
I need help proving two things:
1> that if ? is in Sn, show that ???^-1 is also a cycle of length r.
So far, I know that if ? = (a1 a2 … ar), then I should show that ???^-1 = (?a1 ?a2 … ?ar)
2> if ? is in Sn, show that ? and ???^-1 have the same cycle structure.
(All I know when they say “cycle structure”, is they mean that in the factorization into disjoint cycles have the same number of cycles of each length. Do I need to know something else, too?)
Thanks for any help again.