# Power of a cycle

let π be a product of disjoint m-cycles. Prove that π is a power of a cycle?

So this is like asking show that π = βx 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.

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

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.

micromass
Staff Emeritus
Homework Helper
First, note that

$$\sigma (i_1~i_2~i_3~...~i_n) \sigma^{-1}=(\sigma(i_1)~\sigma(i_2)~\sigma(i_3)~...~\sigma(i_n))$$

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.

let ∏= ∏αi for all 1≤i≤n with αi = (ai1 ai2 ai3 ...aim)

consider θ = (a11 a21......an1 a12 a22......a1m a2m a3m...anm)

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

Hence ∏ = θn