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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, p-groups 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 p-subgroups, Sylow’s three theorems. Read : GT [58-62] and GT [68-75] 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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