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 de±ned 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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This note was uploaded on 10/20/2009 for the course MATH 302 taught by Professor Edwards during the Spring '03 term at CSU Fullerton.
 Spring '03
 Edwards
 Algebra

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