TITLE: Number Theory: confusion with notation
QUESTION [12 upvotes]: In elementary number theory, I find the following notations being used interchangeably, which leads me to have many ambiguous assumptions:

$\mathbb{Z}_p^\times$;  
$\mathbb{Z}_p$,  

where p is any integer. What's the difference between them?
One more question to add to the kitty: what do they mean when they say "a subgroup of $\mathbb{Z}_p$, where p is prime"?
It will be nice if someone can enlighten me on this elementary notation.
Thanks.

REPLY [10 votes]: $\mathbb{Z}_p$ is, in my opinion, problematic notion. In slightly less elementary number theory it refers to the p-adic numbers, which are very different from the integers $\bmod p$. An unambiguous, if slightly more cumbersome, notation would be $\mathbb{Z}/(p)$ or $\mathbb{Z}/p\mathbb{Z}$.