Number Theory: confusion with notation 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.
 A: Usually the notation $\mathbb{Z}_p$ or $\mathbb{Z}/p\mathbb{Z}$ mean the integers modulo $p$, that is $\{0,\ldots,p-1\}$ where you add and multiply as usually and then take the reminder modulo $p$.
The notation of $\mathbb{Z}_p^\times$ is for those numbers which have a multiplicative inverse modulo $p$, namely all $n$ such that exists $m$ such that $n\cdot m$ is $1$ modulo $p$. These are the numbers coprime to $p$, the greatest common divisor of them and $p$ is $1$.
A subgroup of $\mathbb{Z}_p^\times$ is a subset of these numbers which is closed under multiplication (but not necessarily addition), and every number in this subset also has its multiplicative inverse in there.
For example,
$\mathbb{Z}_5 = \{0,1,2,3,4\}$
$\mathbb{Z}_5^\times = \{1,2,3,4\}$
$\{1,4\}$ is a subgroup of $\mathbb{Z}_5^\times$. Can you see why?
A: $\mathbb{Z}_p$ 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 $\mathbb{Z}/p\mathbb{Z}$. 
