solution7

# solution7 - PMath 336: Introduction to Group Theory...

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PMath 336: Introduction to Group Theory Exercise set 7 July 11, 2008 Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works. Notation GL n ( S ) : Invertible n × n matrices with coeﬃcients in S , under multiplication D n : The group of symmetries of the regular n -gon S n : The group of permutations of 1 ,...,n Z : The group of integers under addition Z n : The group { 0 ,...,n - 1 } with addition modulo n U n : The group of numbers less than n and prime to n , with multiplication mod n . Q (quaternions): The subgroup of GL 2 ( C ) generated by i = ± 0 i i 0 ² and j = ± 0 1 - 1 0 ² . Questions 1. [12] Write each of the following permutations as a product of disjoint cycles, and as a product of transposi- tions, and compute its order. (a) ± 1 2 3 4 2 4 3 1 ² (b) ± 1 2 3 4 5 6 7 8 9 4 2 5 7 3 6 1 9 8 ² (c) ± 1 2 3 4 5 2 4 5 1 3 ² Solution: (a) (124) = (12)(24). The order is 3. (b) (147)(35)(89) = (14)(47)(35)(89). The order is 6.

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## This note was uploaded on 04/27/2010 for the course PMATH 336 taught by Professor Moshekamensky during the Spring '08 term at Waterloo.

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solution7 - PMath 336: Introduction to Group Theory...

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