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

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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 ] ² , ± [ 1 ] [ 1 ] [ 0 ] [ 1 ] ² , ± [ 1 ] [ 2 ] [ 0 ] [ 1 ] ² where the entries are in Z 3 . Compute the Cayley table of G with the operation matrix modular multiplication. (10 points)
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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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test1 - Math 3124 Tuesday, September 27 First Test. Answer...

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