How does (beta)(beta)^(-1) equal 2(beta)?

[mcwang719]

Hello this problem is from my modern algebra class.

Let (alpha) and (beta) belong to S(sub n). Prove that (beta)(alpha)(beta)^-1 and (alpha) are both even or both odd.

I got the answer but have a quick question. So let beta=m 2 cycles and alpha= n 2 cycles. Then we have 2m+n and m and must be both even or both odd. My question is how does the 2m come from (beta)(beta)^-1. So does (beta)(beta)^-1=2(beta)?

Any help would be great! thanks

 

[stapel]

Where does the exercise state that ßß^-1 equalled 2ß? Or, if you got this from some other source, what source, and how does it relate?

Thank you.

Eliz.

 

[mcwang719]

it doesn't state that it equals 2(beta) i kinda assumed because (beta)(beta)^-1= 2m right??

 

[Unco]

If \(\displaystyle \beta\) is written as a product of 2-cycles, then \(\displaystyle \beta^{-1}\) is just the same 2-cycles, multiplied in the reverse order. So, of course, if \(\displaystyle \beta\) can be written as m 2-cycles, then so can \(\displaystyle \beta^{-1}\).

 

[mcwang719]

Thanks unco!!! you've been very helpful.