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HW2_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY PROBLEM SET #2 Week #1: September 28th - October 4th. Topics : Universal property of quotients, the isomorphism theorems, subgroups of quotients, direct products, structure of finitely generated abelian groups, the dual group of characters, orthogonality relations, group actions, orbits, stabilizers, the class equation, Cauchy’s theorem on existence of elements of prime order. Read : GT [19-26] and GT [50-57] Problems due Friday, October 2nd, at 4:00 pm in Martin Luu’s mailbox: Problem 1 : Show that S n is generated by the tranpositions s i = ( i, i + 1) , and that they satisfy 1 the braid group relations: That is, s i s j = s j s i when | i - j | > 1 , and s i s j s i = s j s i s j when | i - j | = 1 . Problem 2 : Is D 4 isomorphic to Q 8 ? Problem 3 : Show that any subgroup of Z is of the form n Z , for a unique non-negative integer n . Given positive integers m and n , what are the positive generators of the two subgroups m Z + n Z and m Z n Z ?
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