solution8 - PMath 336: Introduction to Group Theory...

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Unformatted text preview: PMath 336: Introduction to Group Theory Exercise set 8 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 { ,...,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 = i i and j = 0 1- 1 0 . Questions 1. Compute the sign of the following permutations (a) [5] 1 2 3 4 5 6 7 8 9 5 2 3 7 4 1 6 9 8 (b) [5] 1 2 3 4 5 6 7 8 9 3 7 5 6 1 9 2 4 8 Solution: (a) The permutation is equal to (15476)(89), so the sign is- 1. (b) The permutation is equal to (135)(27)(4698), so the sign is 1. 2. For each of the following elements, write the corresponding permutation under the Cayley homomorphism into a symmetric group....
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solution8 - PMath 336: Introduction to Group Theory...

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