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Unformatted text preview: G . Prove that HN = { hn : h H,n N } is a subgroup of G . [10 pts] 4. (a) Let n 5. Give an Abelian subgroup of S n of order 5. [5 pts] (b) Write (3541)(6312)1 (52) in both disjoint cycle notation and as a product of 2cycles. [5 pts] (c) Find an element in S 9 of order 12 which does not x any number. [5 pts] (d) Give 2 nondisjoint cycles in S 5 whose product is not a cycle. [5 pts] 5. Let G = ( Z 15 , +) and H G such that H = h 3 i . List the distinct left cosets of H in G . [7 pts] 6. Prove that ( R \ { } , ) is not cyclic (do not use  R  as your argument). [12 pts] 7. Show that the order of an element is isomorphism invariant. (i.e. If : G = H then o ( g ) = o ( ( g ))) [10 pts] 8. Show that if H G with the property that gH = Hg for all g G then H must be normal in G . (note: Hg = { hg : h H } ) [10 pts]...
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 Spring '11
 Johnson
 Math, Algebra

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