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Unformatted text preview: Ph. D. Algebra Preliminary Exam August 23, 2010 Show the work you do to obtain an answer. Give reasons for your answers. There are 10 questions. Do all parts of all questions. Each question is worth 10 points. Q is the field of rational numbers. C is the field of complex numbers. Assume all rings have identity not equal to 0. 1. Prove that there is no simple group of order 42. 2. Let G , H , and K be groups with | G | = 35, | H | = 60, and | K | = 42. Assume there exist group homomorphisms : G H and : G K with ker 6 = G and ker 6 = G . Prove that ker ker consists of one element. 3. Let T : V W be a surjective linear transformation of vector spaces. Let W 1 and W 2 be subspaces of W such that W = W 1 + W 2 . Prove that V = T- 1 ( W 1 ) + T- 1 ( W 2 ). 4. Let G be a group of order 77 acting on a set X with 20 elements. Prove that the action has at least 2 fixed points....
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- Spring '11