Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup of order 30. 200
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- Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.34. Suppose that and are subgroups of the group . Prove that is a subgroup of .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.
- 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?10. Suppose that and are subgroups of the abelian group such that . If is a subgroup of such that , prove that .4. Prove that the special linear group is a normal subgroup of the general linear group .
- 18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.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.