Homework3 - (a) Find | U ( n ) | for n ∈ { 3 , 4 , 5 , 6...

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Homework 3 Math 332, Spring 2010 These problems must be written up in L A T E X, and are due next Thursday, February 18. 1. Recall that the dihedral group D n has n rotations e,r,. ..,r n - 1 and n reflections s,rs,. ..,r n - 1 s , where r and s satisfy the relations r n = e, s 2 = e, and sr = r - 1 s. (a) If n is odd, prove that all the reflections in D n lie in the same conjugacy class. (b) If n is even, prove that there are exactly two conjugacy classes of reflections in D n . 2. If n is a positive integer, let U ( n ) denote the group of units modulo n (see example 11 in section 2 of the textbook). Note that U ( n ) is always an abelian group.
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Unformatted text preview: (a) Find | U ( n ) | for n ∈ { 3 , 4 , 5 , 6 , 7 , 8 , 15 , 24 } . (b) Identify the isomorphism types of U (3), U (4), U (5), U (6), U (7), and U (8). (c) Determine the orders of the elements of U (15) and U (24). Use your answer to show that these two groups are not isomorphic, and that neither group is cyclic. (d) List the subgroups of U (15), and identify the isomorphism type of each subgroup. (e) List the subgroups of U (24), and identify the isomorphism type of each subgroup. 1...
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This document was uploaded on 11/03/2010.

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