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Prove that the set of natural numbers N form a group under the operation of multiplication.
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- 9. Find all homomorphic images of the octic group.Exercises In Exercises, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition that fails to hold. 8. For a fixed positive integer, the set of all complex numbers such that (that is, the set of all roots of), with operation multiplication.Suppose that G is a finite group. Prove that each element of G appears in the multiplication table for G exactly once in each row and exactly once in each column.
- Prove that the set of all complex numbers that have absolute value forms a group with respect to multiplication.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.Exercises In Exercises, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition that fails to hold. 6. The set of all positive rational numbers with operation multiplication.