# oct04 - Compute the left cosets of h ( 1 2 ) ih [ 1 ] i in...

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Math 3124 Tuesday, October 4 October 4 Ungraded Homework Problem 16.1 page 87 Determine the right cosets of h [ 4 ] i in Z 8 . h [ 4 ] i = { [ 0 ] , [ 4 ] } . Now choose some element not in { [ 0 ] , [ 4 ] } , say [1]. Then h [ 4 ] i ⊕ [ 1 ] = { [ 1 ] , [ 5 ] } . In the same fashion, we have h [ 4 ] i ⊕ [ 2 ] = { [ 2 ] , [ 6 ] } and h [ 4 ] i ⊕ [ 3 ] = { [ 3 ] , [ 7 ] } . We have now exhausted all the elements of Z 8 . Therefore there are four right cosets of h [ 4 ] i in Z 8 , namely { [ 0 ] , [ 4 ] } , { [ 1 ] , [ 5 ] } , { [ 2 ] , [ 6 ] } , { [ 3 ] , [ 7 ] } . Problem 16.5 page 87 Determine the right cosets of h ( 1 3 ) i in S 3 . h ( 1 2 3 ) i = { ( 1 ) , ( 1 2 3 ) , ( 1 3 2 ) } . Now choose an element not in h ( 1 2 3 ) i , say (1 2). Then h ( 1 2 3 ) i ( 1 2 ) = { ( 1 2 ) , ( 1 3 ) , ( 2 3 ) } . We have now exhausted all the elements of S 3 , so we have two right cosets, namely { ( 1 ) , ( 1 2 3 ) , ( 1 3 2 ) } , { ( 1 2 ) , ( 1 3 ) , ( 2 3 ) } Problem 16.18 page 88
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Unformatted text preview: Compute the left cosets of h ( 1 2 ) ih [ 1 ] i in S 3 Z 2 . S 3 Z 2 is a group of order 6 * 2 = 12, while h ( 1 2 ) ih [ 1 ] i is a subgroup of order 2 * 2 = 4, so we expect 12 / 4 = 3 left cosets each with 4 elements. Let H = h ( 1 2 ) ih [ 1 ] i . Then the left cosets of H are H = ((1),[0]), ((1 2),[0]), ((1),[1]), ((1 2),[1]) ((1 2 3),[0])H = ((1 2 3),[0]), ((1 3),[0]), ((1 2 3),[1]), ((1 3),[1]) ((2 3),[0])H = ((2 3),[0]), ((1 3 2),[0]), ((2 3),[1]), ((1 3 2),[1])...
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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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