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ahw5 - 4 Problem 16.2 on page 87 Determine the right cosets...

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Math 3124 Thursday, October 13 Fifth Homework Solutions 1. Problem 14.33 on page 81. Prove or give a counterexample. If a group has a subgroup of order n , then it has an element of order n . S 3 is a subgroup of S 3 of order 6. But the elements of S 3 have order 1, 2 or 3, so we have a counterexample. 2. Problem 15.2 on page 84. Prove that h 24 , - 36 , 54 i = h 6 i . 24 ∈ h 6 i because 24 = 4 * 6, - 36 ∈ h 6 i because - 36 = - 6 * 6, and 54 ∈ h 6 i because 54 = 9 * 6. Therefore h 24 , - 36 , 54 i ⊆ h 6 i . Conversely 6 = 24 + 2 * ( - 36 ) + 54, so 6 ∈ h 24 , - 36 , 54 i and we deduce that h 6 i ⊆ h 24 , - 36 , 54 i . The required equality follows. 3. Problem 15.21 on page 84. (a) List the elements of S 3 × Z 2 . (b) List the elements of the cyclic subgroup h (( 1 2 ) , [ 1 ]) i of S 3 × Z 2 . (c) List the elements of the cyclic subgroup h (( 1 2 3 ) , [ 1 ]) i of S 3 × Z 2 . (a) ((1), [0]) ((2 3), [0]) ((3 1), [0]) ((1 2), [0]) ((1 2 3), [0]) ((1 3 2), [0]) ((1), [1]) ((2 3), [1]) ((3 1), [1]) ((1 2), [1]) ((1 2 3), [1]) ((1 3 2), [1]) (b) ((1), [0]), ((1 2), [1]) (c) ((1), [0]), ((1 2 3), [1]), ((1 3 2), [0]), ((1), [1]), ((1 2 3), [0]), ((1 3 2), [1])
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Unformatted text preview: 4. Problem 16.2 on page 87. Determine the right cosets of h [ 3 ] i in Z 12 . { [0], [3], [6], [9] } { [1], [4], [7], [10] } { [2], [5], [8], [11] } 5. Problem 16.17 on page 88. Compute the right cosets of h (( 1 2 ) , [ 1 ]) i in S 3 × Z 2 . Let H = h (( 1 2 ) , [ 1 ]) i . Then H = { (( 1 ) , [ ]) , (( 1 2 ) , [ 1 ]) } , a subgroup of order 2. Since the order of S 3 × Z 2 is 6 * 2 = 12, we expect 12/2 = 6 cosets. The cosets are H (( 1 ) , [ ]) = { (( 1 ) , [ ]) , (( 1 2 ) , [ 1 ]) } H (( 2 3 ) , [ ]) = { (( 2 3 ) , [ ]) , (( 1 2 3 ) , [ 1 ]) } H (( 3 1 ) , [ ]) = { (( 3 1 ) , [ ]) , (( 1 3 2 ) , [ 1 ]) } H (( 1 2 ) , [ ]) = { (( 1 2 ) , [ ]) , (( 1 ) , [ 1 ]) } H (( 1 2 3 ) , [ ]) = { (( 1 2 3 ) , [ ]) , (( 2 3 ) , [ 1 ]) } H (( 1 3 2 ) , [ ]) = { (( 1 3 2 ) , [ ]) , (( 1 3 ) , [ 1 ]) }...
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