Q: 50
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Q: b. Find all abelian groups, up to isomorphism, of order 360.
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Q: b. Find all abelian groups, up to isomorphism, of order 720.
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Q: 1. Let G be a cyclic group of order 6. How many of its elements generate G?
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Q: Q2.6 Question 1f How many Abelian groups (up to isomorphism) are there of order 36? O 2 O 3 O 4
A: Option D is correct answer
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For each of the following values of n, describe all the abelian groups of order n, up to isomorphism.
n = 36
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- 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.For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19
- Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.True or False Label each of the following statements as either true or false. 4. If is an abelian group, then for all in .
- Exercises 10. For each of the following values of, find all subgroups of the cyclic group under addition and state their order. a. b. c. d. e. f.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.