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Unformatted text preview: H of even integers is a subgroup of G . (d) Let G = Z , the set of integers under ordinary integer addition. Let f : G G be dened by f ( a ) = 2 a for each a G . Show that f is an isomorphism. 7. Let G be a group and let H be a subgroup of G . Let N ( H ) be the subset of all elements in G that commute with all elements of H ; that is, N ( H ) = { g G  ( h H ) ( gh = hg ) } . Show that N ( H ) is a subgroup of G . 8. Let U 18 = { 1 , 5 , 7 , 11 , 13 , 17 } be the group of units in Z 18 . (a) Find the order of each element in U 18 . (b) Show that U 18 is cyclic and nd all the generators of U 18 . 9. Prove Theorem 7.8(3): Let G be a group and suppose a G . If a has nite order n , then a k = e if and only if n  k. 10. Suppose G and H are groups and f : G H is a surjective homomorphism. Prove that if G is abelian, then H is also abelian....
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 Spring '03
 Edwards
 Algebra

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