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Unformatted text preview: Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY PROBLEM SET #3 Week #1: October 5th  October 11th. Topics : Linear independence of characters, fixed point lemma, conjugacy classes, centralizers, centers, pgroups are supersolvable, groups of order 2 p are cyclic or dihedral, groups of order p 2 are abelian, permutations groups, cycles, A n is simple for n ≥ 5, Sylow psubgroups, Sylow’s three theorems. Read : GT [5862] and GT [6875] Problems due Friday, October 9th, at 4:00 pm in Martin Luu’s mailbox: • Problem 1 : Show that Q / Z is isomorphic to μ ∞ ( C ) , the group of z ∈ C * such that z n = 1 for some positive integer n . Is it a finitely generated abelian group? What is the torsion subgroup of Q / Z ? • Problem 2 : Let G be a group, and let G der be the subgroup generated by all commutators [ x,y ] = xyx 1 y 1 , for x,y ∈ G . Show that G der is a normal subgroup of G , and that the corresponding quotient group G ab is abelian....
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This note was uploaded on 05/07/2010 for the course MAT 322 at Princeton.
 '09
 CLAUSSORENSEN
 Algebra, Linear Independence

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