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Algebra Jan 2004 - Algebra Qualifying Examination January...

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Algebra Qualifying Examination January, 2004 Directions: 1. Answer all questions. (Total possible is 100 points.) 2. Start each question on a new sheet of paper. 3. Write only on one side of each sheet of paper. Policy on Misprints: The Qualifying Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem in such a way that it becomes trivial. Notes: 1. All rings are unitary. All modules are unitary. 2. is the rationals, the reals, the complexes, and the integers. Questions 1. (10 points) Determine all groups that have exactly 3 subgroups. 2. (15 points) Let Z ( G ) and Inn( G ) denote the center of the group G and the group of inner automorphisms of G respectively. (i) Prove that G/Z ( G ) is isomorphic to Inn( G ). (ii) Suppose that the group Aut( G ) of automorphisms of G is cyclic. Prove that G is abelian.
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