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- 34. Suppose that and are subgroups of the group . Prove that is a subgroup of .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.If a is an element of order m in a group G and ak=e, prove that m divides k.
- Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?31. (See Exercise 30.) Prove that if and are primes and is a nonabelian group of order , then the center of is the trivial subgroup . Exercise 30: 30. Let be a group with center . Prove that if is cyclic, then is abelian.Let be a subgroup of a group with . Prove that if and only if
- Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.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 .Prove that any group with prime order is cyclic.