midterm2

# midterm2 - PMath 336 Introduction to Group Theory Midterm 2...

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PMath 336: Introduction to Group Theory Midterm 2 June 24, 2008 Instructions Solve all five questions. 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. Simple calculators are allowed, but discouraged. No additional material is allowed. The test duration is 50 minutes. Good luck! Notation D n : The group of symmetries of the regular n -gon S n : The group of permutations of 1 , . . . , n R * : The group of non-zero real numbers under multiplication Z n : The group { 0 , . . . , n - 1 } with addition modulo n U n : The group of positive integers less than n and prime to n , with multiplication mod n . Q (quaternions): The subgroup of GL 2 ( C ) generated by i = 0 ı ı 0 and j = 0 1 - 1 0 . The Exam 1. Let G and H be two groups. (a) [10] Write down the definitions of the following notions: Homomorphism from G to H , kernel of a homomorphism, image of a homomorphism (b) [10] Prove that the kernel and the image of a homomorphism are subgroups (c) [10] Prove that a homomorphism is injective if and only if the kernel is trivial.

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