- #1

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## Homework Statement

Prove that if ##\sigma## is the

*m*-cycle ##(a_1 ~a_2~ \dots ~ a_m)##, then for all ##i \in \{1,2, \dots , m \}##, ##\sigma^i (a_k) = a_{k+i}##. Deduce that ##\sigma^m (a_k) = a_k##

## Homework Equations

## The Attempt at a Solution

I will try to do this by induction. Clearly, ##\sigma (a_k) = a_{k+1}##, as that is how ##\sigma## is defined. Now, suppose that for some ##n \in \mathbb{N}## ##\sigma^n (a_k) = a_{k+n}##. Then ##\sigma^{n+1} (a_k) = \sigma( \sigma^n (a_k) ) = \sigma (a_{k+n}) = \sigma_{a + k + 1} ##.

So I feel like I'm done, but I also feel that I've proved this statement for all ##i \in \mathbb{N}##, rather than for all ##i \in \{1,2, \dots , m \}##. How do I modify it to get it correct?