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Elements Of Modern Algebra
- 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.arrow_forward13. Assume that are subgroups of the abelian group . Prove that if and only if is generated byarrow_forwardLet H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.arrow_forward
- If p1,p2,...,pr are distinct primes, prove that any two abelian groups that have order n=p1p2...pr are isomorphic.arrow_forward15. 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 .arrow_forwardProve or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.arrow_forward
- 10. Suppose that and are subgroups of the abelian group such that . If is a subgroup of such that , prove that .arrow_forward43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .arrow_forwardLet 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.arrow_forward
- Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.arrow_forwardShow that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.arrow_forward39. Assume that and are subgroups of the abelian group. Prove that the set of products is a subgroup of.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,