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review3 - MAT 312/AMS 351 Fall 2010 Review for Final...

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MAT 312/AMS 351 Fall 2010 Review for Final Revised 12/9/10 . NOTE: Final is cumulative. Use review sheets 1 and 2 as well as this one. Also review all homework, as well as midterms 1 and 2. § 3.2 Given a permutation π S ( n ), know how to compute its order (Definition 3.16). Understand how to apply Proposiiton 3.21 to calculate the order of a permutation which has been written as a product of disjoint cycles. Exercise 2 p. 83 . Understand what it means for two permutations π 1 , π 2 S ( n ) to be conjugate (bottom of p.79); that congugacy is an equivalence relation on S ( n ); understand what the shape of a permutation is (Definition 3.26) and that a conjugacy class (defined at top of p. 80) is made up of all the permutations with the same shape (Theorem 3.27). Homework 9, Exercise 5 . Understand the sign of a permutation (Definition 3.31; alternate definition in “Notes and Exercises on Permutations and Matrices”). Know that a permutation with sign 1 is odd , and one with sign 1 is even . Understand that every transposition (Definition 3.4) has sign 1. Know how to prove Lemma 3.35 (every cycle is a product of transpositions) and understand why an odd-length cycle is an even permutation, and vice-versa. Be able to apply Theorem 3.36 (if a permutation is written as a product of transpositions, the permutation is odd if the number of those transpositions is odd, and even if that number is even). Understand that the order, the sign and the shape are the same for conjugate permutations.
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