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jmoney90 · #11 November 4th, 2009, 09:42 PM

Quote:

Originally Posted by Drexel28

An isomorphism? I think not. What binary operation were you supposing of defining on $\mathcal{L}\left(H\right),\mathcal{R}\left(H\right)$ ? But a bijection, yes.

So we can conclude that $\text{card }\mathcal{L}\left(H\right)=\text{card }\mathcal{R}\left(H\right)$

To see this another way, you can easily show that $\Psi:H\mapsto Ha$ is a bijection. Or equivalently that Lagranges theorem is equally applicable to right cosets. From this we can gather that

$|H|\left[G:H\right]_{\text{Left}}=|G|=|H|\left[G:H\right]_{\text{Right}}$

which equivalently shows that $\text{card }\mathcal{L}\left(H\right)=\text{card }\mathcal{R}\left(H\right)$ .

Oh, that is clever! I even noticed it was a bijection, just didn't make that connection lol. I guess I was really focused on the idea of an isomorphism