oct25 - Let G be the abelian group, and suppose the...

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Math 3124 Tuesday, October 25 October 25, Ungraded Homework 1. Problem 22.3 on page 114. Construct the table for Z 12 / h [ 4 ] i . Let us write k to indicate the coset to which k belongs. The order of h [ 4 ] i in Z 12 is 3, so | Z 12 / h [ 4 ] i| = 12 / 3 = 4. The elements of Z 12 / h [ 4 ] i are 0 = { [ 0 ] , [ 4 ] , [ 8 ] } , 1 = { [ 1 ] , [ 5 ] , [ 9 ] } , 2 = { [ 2 ] , [ 6 ] , [ 10 ] } , 3 = { [ 3 ] , [ 7 ] , [ 11 ] } . The Cayley table will be 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 2. Problem 22.5 on page 114. Prove that every quotient group of an abelian group is abelian.
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Unformatted text preview: Let G be the abelian group, and suppose the quotient group is G / H where H G . Since G is abelian, xy = yx for all x , y G . Therefore HxHy = Hxy = Hyx = HyHx for all x , y G . Since the general element of G / H is of the form Hx for some x G , it follows that G / H is abelian....
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This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.

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