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This is An Introduction to the Theory of Groups by George Polites, example 11, page 9. Later on he says:

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I say any prime will do. If m is a prime then G will be a field for ordinary sum and product. In a field the elements different from zero are a group and this group is cyclic. I remember it was not a trivial matter to prove that group is cyclic. How can the author put such a difficult problem at such an elementary stage? He has just shown Lagrange theorem.

The fact is I cannot solve the exercise. I chose this book because I hoped it would be easy but maybe I am wrong.