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midterm1

# midterm1 - PMath 336 Introduction to Group Theory Midterm 1...

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PMath 336: Introduction to Group Theory Midterm 1 May 30, 2008 Instructions Write your answers in the notebook provided. Include full arguments in the answer. Don’t forget to write your name and id number on the notebook. No additional material is allowed. The test duration is 50 minutes. The total number of points is 132. To get full marks, solve correctly a proper subset of the questions that adds up to at least 100 (i.e., either solve questions 1–3, and one of 4–6 or solve two of the questions 1–3 and questions 4–6.) You cannot get more than 100. Good luck! Notation GL n ( S ) : Invertible n × n matrices with coefficients in S , with matrix multiplication SL n ( S ) : Subgroup of GL n ( S ) of matrices with determinant 1 C , R , Q , Z : Set (or additive group) of complex, real, rational and integer numbers, respectively. h A i : Subgroup generated by A . The Exam 1. Let G be a group. (a) [12] Show that if G has more than two elements, and a is an element of G , then G - { a } generates G .

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