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## Homework Statement

In the following problems let ##\alpha## be a cycle of length ##s##, and say

##\alpha = (a_1a_2 . . . a_s)##.

5) If ##s## is odd, ##\alpha## is the square of some cycle of length s. (Find it. Hint: Show ##\alpha = \alpha^{s+1}##)

## Homework Equations

## The Attempt at a Solution

I know ##(12345)(12345) = (12345)## and ##(1234)(1234) = (13)(24)##. And I think I can prove this is true for any integer n, such that if a cycle ##\alpha## has an odd length, then squaring it produces another cycle of odd length, and ##\alpha## has an even length, s, then its square is the product of two disjoint cycles, each of length s/2. I guess i've pretty much restated the problem... i'm stuck.

For the hint.. I know ##\alpha## of length ##s## has ##\alpha## distinct powers but I'm not sure how to prove it.

Sorry this is so bare bones, thank you for any help