1. Let f = (x1, x2, ...., xr) in Sn. Show that o(f) = r.
(I have to do this using cycles or transpositions or something, but I don't know how it would work.)
2. Prove that Sn is non-abelian if n is greater than or equal to 3.
To show that you need to show that (identity permuation) and that if . Also it would be helpful to write (you will see why) instead.
First we show that is the identity permuation. This means we need to show for all . We can assume that because otherwise and so there is no need to check that case. Remember that (this is why we rewrote the indicies differently). Therefore, . If we want it must mean that . Therefore, the smallest such is and so .
Compare .2. Prove that Sn is non-abelian if n is greater than or equal to 3.