- #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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In summary, we can prove that a product of disjoint m-cycles is a power of a cycle by showing that it can be written as a product of transpositions and then finding a specific cycle that when raised to a power, gives the original product. This shows that the product is a power of a cycle, as desired.

- #1

- 376

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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

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- #2

Science Advisor

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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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mathwonk said: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

The "power" of a cycle refers to its ability to convert energy from one form to another, typically through the motion of its wheels.

The power of a cycle is typically measured in watts, which is a unit of power equal to one joule of energy per second.

The power of a cycle can be affected by various factors such as the cyclist's strength and technique, the weight and design of the cycle, and the terrain being ridden on.

Yes, the power of a cycle can be increased through various means such as training and improving cycling technique, using a lighter and more aerodynamic cycle, and choosing routes with less resistance.

The power of a cycle is closely related to its speed, as a higher power output from the cyclist can result in a faster speed. However, other factors such as wind resistance and terrain also play a role in determining the speed of a cycle.

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