Hi: This is An Introduction to the Theory of Groups by George Polites, example 11, page 9: please see first attachment below. And now he gives this exercise: please see second attachment below.

I say any prime will do. If m is a prime then G will be a field for

sum and product modulo m. In a field the elements different from zero are

a group for multiplication 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.