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- Use mathematical induction to prove that if a is an element of a group G, then (a1)n=(an)1 for every positive integer n.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.Let be a group of order , where and are distinct prime integers. If has only one subgroup of order and only one subgroup of order , prove that is cyclic.
- 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 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?
- Assume that G is a finite group, and let H be a nonempty subset of G. Prove that H is closed if and only if H is subgroup of G.Let be a subgroup of a group with . Prove that if and only if .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 .