Looking for help to understand example of Group I am looking for someone to help me to understand what is going on in the following example, from Hersteins "Topics in Algebra".
It says, 
Let $G$ be the set of all $2*2$ matrices $$\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$$ where $a,b,c,d$ are integers modulo $p$, $p$ is a prime number such that $ad-bc \neq 0$, with matrix multiplication having its usual definition, then verify that $G$ is a non-abelian finite group.
I think I am just even confused on some definitions. From what I understood previously, to say $a\equiv b \quad mod \quad n$ is to say that $n | a-b$.
So what is it to mean the these entries are integers modulo $p$? Can anyone share some insight about this? Thank you.

$G$ is group because if $x,y \in G$ then $x^{-1} y \in G$ and $G$ is nonabelian because:
$\begin{pmatrix} 1 & 0 \\ 1 & 1 \\ \end{pmatrix} *\begin{pmatrix} 0 & 1 \\ 1 & 1 \\ \end{pmatrix} \neq \begin{pmatrix} 0 & 1 \\ 1 &  1\\ \end{pmatrix} *\begin{pmatrix} 1 & 0 \\ 1 & 1 \\ \end{pmatrix}$
And at last $G$ is finite because $|G|= (p^2-1)(p^2-p)$
Remarks: addition and multiplication in $G$ is like this:
For example if $p=7$ then $4*5 + 3^2=6+2=1$ ! because $4*5 \equiv 6 \mod {7}$ and $3^2=9 \equiv 2 \mod {7}$ and then $6+2=8 \equiv 1 \mod {7}$ hence:
$\begin{pmatrix} 4 & 3 \\ 0 & 1 \\ \end{pmatrix} *\begin{pmatrix} 5 & 1 \\ 3 & 0 \\ \end{pmatrix}= \begin{pmatrix} 1 & 4 \\ 3 & 0 \\ \end{pmatrix}$
