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HW3_322_F09

# HW3_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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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. Describe the group D ab n up to isomorphism, for all integers n 3 .

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