It is currently Fri Jan 18, 2019 4:54 pm


All times are UTC [ DST ]




Post new topic Reply to topic  [ 2 posts ] 
Author Message
 Post subject: Even permutation
PostPosted: Tue Oct 11, 2016 9:59 am 

Joined: Sat Nov 14, 2015 6:32 am
Posts: 156
Location: Melbourne, Australia
Let $\alpha$ and $\beta$ be elements of $\mathcal{S}_n$. Prove that ${\alpha}^{-1}{\beta}^{-1}\alpha \beta$ is an even permutation.

_________________
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$


Top
Offline Profile  
Reply with quote  

 Post subject: Re: Even permutation
PostPosted: Tue Oct 11, 2016 2:13 pm 
Team Member

Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
Let \(\displaystyle{\sigma:S_{n}\to \mathbb{Z}_{2}}\) be the sign map, which is ring epimorphism.

Then,

\(\displaystyle{\begin{aligned} \sigma(a^{-1}\,\beta^{-1}\,a\,\beta)&=-\sigma(a)-\sigma(b)+\sigma(a)+\sigma(b)\\&=0\end{aligned}}\)

so, the permutation \(\displaystyle{a^{-1}\,\beta\,a\,\beta}\) is even.


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 2 posts ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
cron
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net