Behaviors of Z modulo

Short, sweet, to the point.

Behavior of ${\mathbb{Z}}_p$ for prime P.

1. It is a field so everything in Fields. All finite integral domains are fields.
   1. Every non-zero element has inverse
   2. For every $ab=0$ either $a=0$ or $b=0$

Behavior of ${\mathbb{Z}}_n$

1. Is only a unit iff $GCD(a,n)=1$ (relatively prime / share no factors)
2. As a group, the generators of ${\mathbb{Z}}_n$ are the elements which are relatively prime to n. This is because the orders of the subgroups must divide the parent group according to langrage.

Important for order of elements in ${\mathbb{Z}}_n$

Taken from Order of elements in Cyclic Groups

Let $G$ be a cyclic group of order $n$ and $a$ generate $G$, ie $\mid ⟨a⟩\mid =n$. Then for $b=a^k\in ⟨a⟩$, order of b is $\frac{n}{gcd(n,k)}$

Practice:

In ${\mathbb{Z}}_{12}$ through this we know the order of 1 is $\frac{12}{gcd(1,12)}=12$. order of 2 is 6 order of 5 is also 12 so $⟨5⟩=⟨1⟩$ This adds more intuition to what we proved in All Ideals in integers mod n.