Afternoon guys, I'm stuck on this proof and I'd really appreciate it if someone could help solve it. ( = set of integers)

Question/Theorem

Let be the set of nonzero integers modulo p. ( ) Prove is a group with multiplication mod if and only if is a prime number.

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My (lame) effort so far:

We need to show:

(1) is a group with multiplication mod is prime.

(2) if is prime, is a group with multiplication mod

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(1) , such that

so

p divides

such that

I don't know where to go from here, or indeed if this is the right way of approaching this.

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(2) if is prime, then any integer such that is coprime to and so n,m such that

Now using that, we need to show the properties of a group are satisfied.

Showing associativity and identity (=1) are easy enough. But I don't know how to show an inverse exists. i.e. show that such that using the above property of a prime number.

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Please help

Thank you.