st2 - θ 1 = 1 2 3 4 5(don’t verify Determine θ 9(6...

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Math 3124 Thursday, October 20 Sample Second Test. Answer All Problems. Please Give Explanations For Your Answers. 1. Show that ( 1 2 ) , ( 1 2 3 4 ) = ( 2 3 4 ) , ( 1 3 4 2 ) . (6 points) 2. The following is part of a Cayley table for a group. Complete the table. a b c d a b c c b d (i.e. given bc = c and cd = b .) (6 points) 3. Determine the right cosets of ([ 1 ] , ( 1 2 )) in Z 3 × S 3 . (6 points) 4. Let A , B G . Suppose | G | = 49, G = A , G = B , and A = B . Prove that A B = { e } . (6 points) 5. Determine the number of subgroups of a cyclic group of order 720. Also give an example of a group of order 720 which has at least 15 subgroups of order 2. (6 points) 6. Let p and q be primes (not necessarily distinct) and let G be a group of order pq . Prove that every proper subgroup of G (i.e. every subgroup apart from G itself) is cyclic. Deduce that G has at most pq + 1 subgroups. (6 points) 7. Explain why S 3 × S 5 S 6 . (6 points) 8. There is a unique homomorphism θ : Z
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Unformatted text preview: θ ([ 1 ]) = ( 1 2 3 )( 4 5 ) (don’t verify). Determine θ ([ 9 ]) . (6 points) 9. Let H = h ( 1 2 3 ) , ( 1 2 )( 3 4 ) i . (a) Prove that h ( 1 2 3 ) i is not a normal subgroup of H . (b) Prove that | H | 6 = 6. (c) Prove that H ± S 4 . (6 points) 10. Let H ± G be groups and suppose every nonidentity element of G has order 5. Prove that every nonidentity element of G / H has order 5. (6 points) 11. Let G = Z 4 × Z 4 and let H = h [ 2 ] i×h [ 2 ] i . (a) Write down the right cosets of H in G . (b) Construct the Cayley table for G / H . (6 points) Test on Tuesday, November 1. Material sections 15–22 excluding 20 approximately . RSA encryption will not be examined. Review session on Sunday October 30 at 4:50 p.m. in McBryde 126....
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