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test2 - Math 4124 Monday April 4 Second Test Answer All...

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Math 4124 Monday, April 4 Second Test. Answer All Problems. Please Give Explanations For Your Answers 1. Let G be a group and suppose there is a homomorphism of G onto Z / 2 Z × Z / 2 Z with kernel K . Determine the number of subgroups of G which contain K . (12 points) 2. Compute the orders of the conjugacy class and the centralizer of (1 2)(3 4)(5 6)(7 8 9) in S 9 (you can leave your answers as a product of numbers). (12 points) 3. Let p be a prime and let G be a nonabelian group of order p 3 . Prove that G / Z ( G ) = Z / p Z × Z / p Z . (12 points) 4. Prove that a group of order 45 is abelian. (12 points) 5. Prove that a group of order 1452 = 2 2 · 3 · 11 2 is not simple. (12 points)
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Math 4124 Monday, April 4 Solutions to Second Test 1. By the fundamental homomorphism theorem, G / K = Z / 2 Z × Z / 2 Z . From the sub- group correspondence theorem, the number of subgroups of G containing K equals the number of subgroups of G / K . Therefore we need to determine the number of subgroups of Z / 2 Z × Z / 2 Z . Two of the subgroups are 1 and Z / 2 Z × Z / 2 Z . All other
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