Lets suppose that X & Z are members of some arbitrary $S_n$ group and that X & Z are cycles. Then we define cycles X & Z by:
$ X = (a_1 a_2 ... a_m) $ - which are pairwise distinct members of $\mathbb Z^+$
$ Z = (b_1 b_2 ... b_n) $ - which are pairwise distinct members of $\mathbb Z^+$
-- Suppose that $ X \circ Z = Z \circ X $ , then must it be true that $ a_1, ..., a_m, b_1, ..., b_n $ are pairwise distinct?
Intuitively I don't think that they "must" be pairwise distinct since there could be overlap between what elements are in X & Z while they still satisfy their own constraint of being individually pairwise distinct
For X = (1 5 8 6 4 ) and Z = (2 3 7 9 1) are pairwise distinct in their own cycle, but when conjoined through their operation of composition they are no longer pairwise distinct since there is a repeating '1'. But I am having trouble formulating this into a formal proof that expresses this idea (assuming it is correct)
any and all help is appreciated