If the generator of a cyclic group G is of infinite order (or of zero order), then G is isomorphic to the additive group of integers.
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If the generator of a cyclic group G is of infinite
order (or of zero order), then G is isomorphic to
the additive group of integers.
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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.Exercises 35. Prove that any two groups of order are isomorphic.Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.
- 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.Label each of the following statements as either true or false. Any two cyclic groups of the same order are isomorphic.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.
- 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 .Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.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.
- Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.True or False Label each of the following statements as either true or false. The order of an element of a finite group divides the order of the group.