103a_07w_fe

103a_07w_fe - Math 103A Final Exam (100 points) 11:30AM...

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Math 103A Final Exam (100 points) 11:30AM Monday 3/19/2007 Please put your name and ID number on your blue book. CLOSED BOOK, but BOTH SIDES of two pages of notes are allowed. Calculators are NOT allowed. In a multipart problem, you can do later parts without doing earlier ones. You must show your work to receive credit. 1. (12 pts.) Which are TRUE and which are FALSE? Do NOT give reasons. (a) If ϕ : G H is a group homomorphism and K ⊳ H , then ϕ 1 ( K ) ⊳ G . (b) If ϕ : G H is a group homomorphism and H is a cyclic group, then G is a cyclic group. (c) If H ⊳ G and K ⊳ H , then K ⊳ G . (d) Suppose p and q are primes. Then Z pq Z p Z q if and only if p n = q . (e) If G is a group of order n and k divides n , then G has an element of order k . (f) If g G , then the order of g divides the order of G . 2. (12 pts.) Let G be a group and Z ( G ) its center; that is, Z ( G ) = { g G | xg = gx for all x G } . (a) Prove that
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103a_07w_fe - Math 103A Final Exam (100 points) 11:30AM...

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