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test1 - Math 3124 Tuesday September 27 First Test Answer...

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Math 3124 Tuesday, September 27 First Test. Answer All Problems. Please Give Explanations For Your Answers. 1. Which of the following formulae for x * y defines an associative operation on Z # 4 , the set { [ 1 ] , [ 2 ] , [ 3 ] } . (a) x * y = x y (b) x * y = y (10 points) 2. Let α = (4 6 7) - 1 (3 4)(1 2 3 6 5). Write α as a product of disjoint cycles. Also determine whether α A 7 . (10 points) 3. Let G = S 4 and T = { 1 , 3 } . Write all the elements of G T and G ( T ) , as a product of disjoint cycles. (Recall that G T = { α G | α ( t ) = t for all t T } and G ( T ) = { α G | α ( T ) = T } .) (10 points) 4. Let G be a group, let X G , and let C = { g G | gx = xg for all x X } . Prove that C is a subgroup of G . (10 points) 5. Let G consist of the three matrices [ 1 ] [ 0 ] [ 0 ] [ 1 ] , [ 1 ] [ 1 ] [ 0 ] [ 1 ] , [ 1 ] [ 2 ] [ 0 ] [ 1 ] where the entries are in Z 3 . Compute the Cayley table of G with the operation matrix modular multiplication. (10 points) 6. Compute the order of ( ( 1 2 3 4 )( 5 6 7 8 9 10 ) , [ 1 ] ) in S 10 × ( Z 10 , ) . (10 points)
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Math 3124 Tuesday, September 27 First Test Solutions 1. (a) This is not an associative operation, in fact it is not even an operation. For example
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