Consider the group G={x ER such that x#0} under the binary operation *: The identity element of G is e=-1/2. The order of the element +2 is: * 00 3 O 2 8. о
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A: Thanks for the question :)And your upvote will be really appreciable ;)
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Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.
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A: Thanks for the question :)And your upvote will be really appreciable ;)
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A: Thanks for the question :)And your upvote will be really appreciable ;)
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- Consider the group U9 of all units in 9. Given that U9 is a cyclic group under multiplication, find all subgroups of U9.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
- 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .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.Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .
- Exercises 27. Consider the additive groups , , and . Prove that is isomorphic to .16. Suppose that is an abelian group with respect to addition, with identity element Define a multiplication in by for all . Show that forms a ring with respect to these operations.Prove that if r and s are relatively prime positive integers, then any cyclic group of order rs is the direct sum of a cyclic group of order r and a cyclic group of order s.