test1 - Math 5125 Wednesday, September 28 First Test....

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Math 5125 Wednesday, September 28 First Test. Answer All Problems. Please Give Explanations For Your Answers 1. Let H be a subgroup of the finite group G , and let p be a prime. Prove that two distinct Sylow p -subgroups of H cannot be contained in the same p -subgroup of G . (16 points) 2. Prove that a group of order 280 cannot be simple. (17 points) 3. Let A and B be finitely generated abelian groups. Prove that if A × A = B × B , then A = B . (17 points)
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Math 5125 Wednesday, September 28 Solutions to First Test 1. Let P 1 , P 2 be Sylow p -subgroups of H , and suppose Q is a p -subgroup of G containing P 1 and P 2 . Since Q H is a p -group containing P 1 and P 1 is a Sylow p -subgroup, we must have Q H = P 1 . Similarly Q H = P 2 and we conclude that P 1 = P 2 , as required. 2. Let G be a simple group of order 280 = 8 * 5 * 7. The number of Sylow 2-subgroups is congruent to 1 mod 7 and divides 40. This number cannot be 1, because then G would have a normal subgroup of order 8. Therefore
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test1 - Math 5125 Wednesday, September 28 First Test....

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