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homework7

# homework7 - Math 113 Section 5 Fall 2010 Homework#7 Due...

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Math 113 Section 5 Homework #7 Fall 2010 Due: Monday, October 25 1. Section 4.3, Exercise 5: Let G be a group. If the center of G is of index n , prove that every conjugacy class has at most n elements. 2. Section 5.1, Exercise 1: Let G 1 , G 2 , . . . , G n be groups. Show that the center of a direct product is the direct product of the centers: Z ( G 1 × G 2 × · · · × G n ) = Z ( G 1 ) × Z ( G 2 ) × · · · × Z ( G n ) . Deduce that a direct product of groups is abelian if and only if each of the factors is abelian. 3. Section 5.1, Exercise 2: Let G 1 , G 2 , . . . , G n be groups and let G = G 1 × · · · × G n . Let I be a proper, nonempty subset of { 1 , . . . , n } and ket J = { 1 , . . . , n } - I . Define G I to be the set of elements of G that have the identity of G j in position j for all j J . (a) Prove that G I is isomorphic to the direct product of the groups G i , i I . (b) Prove that G I is a normal subgroup of G and G/G I is isomorphic to G J . (c) Prove that G is isomorphic to G I × G J .

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homework7 - Math 113 Section 5 Fall 2010 Homework#7 Due...

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