Find any case in which the number of subgroups with an order of 3 can be exactly 4 in the Abelian group with an order of 108.
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- 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.Exercise 8 states that every subgroup of an abelian group is normal. Give an example of a nonabelian group for which every subgroup is normal. Exercise 8: Show that every subgroup of an abelian group is normal.4. List all the elements of the subgroupin the group under addition, and state its order.
- 12. Find all normal subgroups of the quaternion group.Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.10. Prove that in Theorem , the solutions to the equations and are actually unique. Theorem 3.5: Equivalent Conditions for a Group Let be a nonempty set that is closed under an associative binary operation called multiplication. Then is a group if and only if the equations and have solutions and in for all choices of and in .