- #1

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So this is like asking show that π = β

^{x}for some cycle β and pos. integer x. right?

I don't know how to proceed on this except for the fact that the order of π is m.

any hints please

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- Thread starter Bachelier
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- #1

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So this is like asking show that π = β

I don't know how to proceed on this except for the fact that the order of π is m.

any hints please

- #2

mathwonk

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what have you tried? remember, when you have no clue what will work, any idea at all is progress.

- #3

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what have you tried? remember, when you have no clue what will work, any idea at all is progress.

man I have no idea.

I know I can write π = (....)(....)(....)(....)(....)(....)(....)(....)(....)(....)

probably need to consider when m is even and odd.

I can break down π to a product of transpositions.

But the end result is too abstract I can get my head around it.

- #4

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[tex]\sigma (i_1~i_2~i_3~...~i_n) \sigma^{-1}=(\sigma(i_1)~\sigma(i_2)~\sigma(i_3)~...~\sigma(i_n))[/tex]

This allows you to bring everything back to the cycle (1 2 3 ... n).

Now, take powers of (1 2 3 ... n) and see what types of disjoint cycle decompostions you meet.

- #5

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consider θ = (a

then applying θ n times will give us the original ∏.

Hence ∏ = θ

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