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- Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?Show that a group of order 4 either is cyclic or is isomorphic to the Klein four group e,a,b,ab=ba.Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
- Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.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.Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
- 10. Suppose that and are subgroups of the abelian group such that . If is a subgroup of such that , prove that .18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.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.