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Unformatted text preview: Math 3124 Tuesday, November 1 Second Test. Answer All Problems. Please Give Explanations For Your Answers. 1. In Z 24 , prove that [ 4 ] , [ 6 ] = [ 10 ] . (10 points) 2. The following is part of the Cayley table for a group. Complete the table. a b c d a b b c b d (i.e. given a 2 = cd = b .) (10 points) 3. Let G be a group of order 25 and let A , B be subgroups of G with  A  =  B  . Prove that either A = B or A ∩ B = { e } . (10 points) 4. Let G be a cyclic group of order 72 and let H be a subgroup of order 2. Determine the number of subgroups of G containing H (including G and H ). 5. Prove that S 5 S 4 × Z 5 . (10 points) 6. Let G = Z 12 and let H = [ 3 ] . (a) Write down the right cosets of H in G . (b) Construct the Cayley table for G / H . (10 points) Math 3124 Tuesday, November 1 Second Test Solutions 1. Since [ 10 ] = [ 4 ] ⊕ [ 6 ] we see that [ 10 ] ⊆ [ 4 ] , [ 6 ] . Now [ 4 ] = [ 10 ] ⊕ [ 10 ] and [ 6 ] = [ 10 ] ⊕ [ 10 ] ⊕ [ 10 ] and we deduce that [ 4 ] , [...
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 Fall '08
 PARRY
 Math, Algebra, Group Theory, Cyclic group, Coset, Cayley table

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