# Algebra2007Jan - Ph D Algebra Preliminary Exam Wednesday...

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Ph. D. Algebra Preliminary Exam Wednesday, January 10, 2007 Show the work you do to obtain an answer. Give reasons for your answers. There are 10 questions. Do all parts of all questions. Notations: C ( x ) is the centralizer of a group element x , , ·i is the usual inner product and k · k is the usual metric on R n , A T is the transpose of a matrix A , I n is the n × n identity matrix. 1. (10 points) Let G be a ﬁnite group having exactly one nontrivial proper subgroup. Prove that G is cyclic of order p 2 for some prime number p . 2. Let G be a ﬁnite group having n distinct conjugacy classes. (a) (8 points) Prove the identity x G | C ( x ) | = n | G | . (b) (2 points) Compute the probability that two randomly chosen elements of G commute. The random selection is done ”with replacement” so that choosing the same element twice is a possible outcome. 3. (10 points) Let A be an n × n real matrix, and prove that the following (criteria for A to be orthogonal) are equivalent. (a)

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Algebra2007Jan - Ph D Algebra Preliminary Exam Wednesday...

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