Afternoon guys, I'm stuck on this proof and I'd really appreciate it if someone could help solve it. ( = set of integers)
Let be the set of nonzero integers modulo p. ( ) Prove is a group with multiplication mod if and only if is a prime number.
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
(1) , such that
I don't know where to go from here, or indeed if this is the right way of approaching this.
(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.