Unformatted text preview: and O for the coset O D .12/ E D E .12/ consisting of odd permutations. The subgroup E is the kernel of the sign homomorphism ± W S n ²! f 1; ² 1 g . Since the product of two permutations is even if, and only if, both are even or both are odd, we have the following multiplication table for the two cosets of E : E O E E O O O E The products are taken in the sense mentioned previously; namely the product of two even permutations or two odd permutations is even, and the product of an even permutation with an odd permutation is odd. Thus the multiplication on the cosets of E reproduces the multiplication on the group f 1; ² 1 g . This is a general phenomenon: If N is a normal subgroup of a group G , then the set G=N of left cosets of a N in G has the structure of a group....
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 Fall '08
 EVERAGE
 Algebra, Group Theory, conjugacy classes, Homomorphism Theorems

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