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.
Now I'm going to be very, very cheeky and ask for any points for the 2nd bit of the proof. i.e. we can't assume p is prime, and deduce it is from the fact that satisying the properties of a group.
I've outlined my thoughts in my original post, but admittedly, I don't have any concrete ideas. If you, or anyone else, can nudge me in the right direction, that'd be superb.
Suppose (nobody ever cares for when because the group on only one element is really unintersing and never comes up anyway). We will show is not a field. If it is a field then if (why?). But if is not a prime then we can write where , thus while . This means it cannot be a field if has factors.