sep29 - Determine the elements in each of the following...

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Math 3124 Thursday, September 29 September 29 Ungraded Homework Problem 14.4 page 79 Determine the elements in each of the cyclic subgroups of Z 6 . Also give the order of each element of Z 6 . h [ 0 ] i [0] h [ 1 ] i [0], [1], [2], [3], [4], [5] h [ 2 ] i [0], [2], [4] h [ 3 ] i [0], [3] These are all the cyclic subgroups (note h [ 4 ] i = h [ 2 ] i etc.). The order of [0] is 1, of [1] is 6, of [2] is 3, of [3] is 2, of [4] is 3, of [5] is 6. Problem 14.5 page 79 Find the order of (1 2)(3 4) in S 4 . Verify that { (1),(1 2),(3 4),(1 2)(3 4) } is an Abelian noncyclic subgroup of S 4 . Since (1 2)(3 4) is already written as a product of disjoint cycles, the order is [2,2] = 2. To verify that is an abelian subgroup of S 4 , just check closure, commutativity, inverses (here each element is its own inverse) and identity. Finally if this abelian subgroup was a cyclic subgroup, then because the order of the subgroup is 4, it would have an element of order 4. However all the three nonidentity elements have order 2, so the subgroup is not cyclic. Problem 14.8 page 79
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Unformatted text preview: Determine the elements in each of the following subgroups of the group of symmetries of a square (Table 8.1). (a) h 90 i (b) h 180 i (c) h 270 i (a) 2 90 = 180 , 3 90 = 90 180 = 270 , 4 90 = 90 270 = . Therefore 90 has order 4 and we see that h 90 i = { , 90 , 180 , 270 } . (b) 2 180 = . Therefore 180 has order 2 and we see that h 180 i = { , 180 } . (c) 2 270 = 180 , 3 270 = 270 180 = 90 , 4 270 = 270 90 = . Therefore 270 has order 4 and we see that h 270 i = { , 90 , 180 , 270 } . Problem 15.22 page 85 (a) List the elements in the subgroup h ([ 2 ] , [ 2 ]) i of Z 4 Z 8 . (b) List the elements in the subgroup h [ 2 ] ih [ 2 ] i of Z 4 Z 8 . (a) ([0],[0]), ([2],[2]), ([0],[4]), ([2],[6]) (b) ([0],[0]) ([0],[2]) ([0],[4]) ([0],[6]) ([2],[0]) ([2],[2]) ([2],[4]) ([2],[6])...
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