midterm3 - PMath 336: Introduction to Group Theory Midterm...

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PMath 336: Introduction to Group Theory Midterm 3 July 14, 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 other material is allowed. The test duration is 50 minutes. Good luck! Notation S n : The group of permutations of 1 ,...,n A n : The even permutations in S n U n : The group of positive integers less than n and prime to n , with multiplication mod n . The Exam 1. Let α = ± 1 2 3 4 5 6 7 8 9 7 4 3 2 9 1 6 5 8 ² β = ± 1 2 3 4 5 6 7 8 9 8 5 2 1 4 7 3 6 9 ² For each of α , β and αβ : (a) [16] Write it as a product of disjoint cycles (b) [16] Write it as a product of transposition (c) [14] compute the order and the sign Solution: (a) α = (176)(598)(24), β = (18673254), αβ = (15298)(347) (b) α = (17)(76)(59)(98)(24) β = (18)(86)(67)(73)(32)(25)(54) αβ = (17)(76)(59)(98)(24)(18)(86)(67)(73)(32)(25)(54)
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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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midterm3 - PMath 336: Introduction to Group Theory Midterm...

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