Suppose G = (a) is a cyclic group of order 6. Find all the subgroups of G and list the elements in each of these subgroups.
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- 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.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.Exercises 35. Prove that any two groups of order are isomorphic.
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Label each of the following statements as either true or false. Any two cyclic groups of the same order are isomorphic.If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.
- Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.Find all subgroups of the octic group D4.15. 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 .