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Unformatted text preview: Modern Algebra 1
Spring 2010 Homework 2.1
Due January 27 Exercise 1. Let
σ= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
13 2 15 14 10 6 12 3 4 1 7 9 5 11 8 τ= 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
14 9 10 2 12 6 5 11 15 3 8 7 4 1 13 and let
. Find the cycle decompositions of each of the following permutations: σ , τ , στ , τ σ , and τ 2 σ . Exercise 2. For each of the permutations α whose cycle decompositions you computed in
the preceding exercise, determine the least n ∈ N so that αn = (1).
Exercise 3.
a. If τ = (1 2)(3 4)(5 6)(7 8)(9 10) determine whether there is an ncycle σ (n ≥ 10) with
τ = σ k for some integer k .
b. If τ = (1 2)(3 4 5) determine whether there is an ncycle σ (n ≥ 5) with τ = σ k for some
integer k . Exercise 4. Prove that Sym(N) is inﬁnite (do not say that ∞! = ∞). ...
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This note was uploaded on 02/29/2012 for the course MATH 3362 taught by Professor Ryandaileda during the Spring '10 term at Trinity University.
 Spring '10
 RyanDaileda
 Algebra

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