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Unformatted text preview: S UMMARY OF 10/27 L ECTURE Began by giving an intuition to the quotient group G/H : think of G as the body, and G/H as an X-ray. Many aspects of G are preserved in G/H , and even if you’re just given G/H you can partially reconstruct some features of G . But you’re losing some data – depth – when you go from the 3 dimensional body, to the two dimensional representation of it (the X-ray). The last two lectures, we’ve discussed how, given H ≤ G with G abelian, one can form the quotient group G/H which is the set of all translations of H (called cosets ): G/H = n gH o g ∈ G under the binary operation inherited from G : ( aH )( bH ) = ( ab ) H. What prevents us from making the same construction of G/H for G non-abelian? Well, noth- ing. In fact, we constructed the set G/H during the course of our proof of Lagrange’s theorem, and there was no requirement that G be abelian. The difficulty arises if you want G/H to be a group . Here’s why....
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This note was uploaded on 03/26/2011 for the course MAT 301 taught by Professor Gideonmaschler during the Fall '10 term at University of Toronto- Toronto.
- Fall '10