Proving the Power of a Cycle in Disjoint m-Cycles: A Comprehensive Guide

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
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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.

any hints please
 
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  • #2
what have you tried? remember, when you have no clue what will work, any idea at all is progress.
 
  • #3
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
First, note that

[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
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
 

1. What is the "power" of a cycle?

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

2. How is the power of a cycle measured?

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

3. What factors affect the power of a cycle?

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.

4. Can the power of a cycle be increased?

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.

5. How does the power of a cycle relate to its speed?

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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