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- Let G be an abelian group. For a fixed positive integer n, let Gn={ aGa=xnforsomexG }. Prove that Gn is a subgroup of G.6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
- 9. Let be a group of all nonzero real numbers under multiplication. Find a subset of that is closed under multiplication but is not a subgroup of .Let be a subgroup of a group with . Prove that if and only if .13. Let be an abelian group with respect to multiplication. Prove that each of the following subsets of is subgroup of. a. b. for a fixed positive integer .
- 5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:44. Let be a subgroup of a group .For, define the relation by if and only if . Prove that is an equivalence relation on . Let . Find , the equivalence class containing .Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.