Show that the nonzero elements in Z_n form a group under the product [a][b]=[ab] if and only if n is a prime. I know that they definitely do not form a group when n is not prime but I am not sure how to prove otherwise. Please show steps. Thank you!
Show that the nonzero elements in Z_n form a group under the product [a][b]=[ab] if and only if n is a prime. I know that they definitely do not form a group when n is not prime but I am not sure how to prove otherwise. Please show steps. Thank you!
What you are trying to show is that $\displaystyle \mathbb{Z}_p^{\times}$ is a group.
Multiplication is well defined and that is pretty easy to show. just show that $\displaystyle (a+pz)(b+py)=ab+apy+bpz+p^2yz = ab + p( ay + bz + pyz) \equiv ab (mod p)$
so it is closed under multiplication
You just gotta show it has an inverse. But this too is easy, if $\displaystyle 0 \not = a\in \mathbb{Z}_p$, then you know p and a are relatively prime. Otherwise p would not be prime as a and p would share a divisor.
but this means there exist integers x and y so that
$\displaystyle ax+py=1$ but this tells you when you reduce mod p
$\displaystyle ax \equiv 1 (mod p)$
so x reduced mod p is the inverse of a. Thus you have shown it to be a group