Power of a cycle

  • Thread starter Bachelier
  • Start date
  • #1
376
0

Main Question or Discussion Point

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
 

Answers and Replies

  • #2
mathwonk
Science Advisor
Homework Helper
10,822
987
what have you tried? remember, when you have no clue what will work, any idea at all is progress.
 
  • #3
376
0
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
22,097
3,279
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
376
0
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
 

Related Threads on Power of a cycle

  • Last Post
Replies
1
Views
2K
Replies
8
Views
6K
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
2K
Replies
3
Views
2K
  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
13
Views
5K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
9
Views
3K
Top