group

After we choose two elements from a set $G$ ,
let us make a new set
\[ G\times G=\left\{(a,b) ; a,b\in G \right\} .  \]

This set $G\times G$  is called a direct product or a cartesian product of the set $G$ .
In general, the cartesian product of $G$ and $H$ is a set of ordered pairs of two different sets.
\[  \left\{ \lt a,b\gt ; a\in G, b\in H  \right\}  \]

There is a mapping or function $\rho$ from the set $G\times G$ to the set $G$.
\[ \rho : G\times G \rightarrow G  \]
Thus,
\[ \rho((a,b))=x\in G  \]
or as the mapping $\rho$ is a binary operation,
\[  a\rho b=x  \]

If the operation $\rho$ from $G\times G$ to $G$ satisfies the below condition,

(1)$a\rho(b\rho c)=(a\rho b)\rho c\qquad (a,b,c\in G)$
(2)for any element $a$ in $G$ , there exists a element $e\in G$ such that $a\rho e=a$ and $e\rho a=a$ .
(3)for any element $a$ in $G$ , there exists a element $y\in G$ such that $a\rho y=y\rho a=e$ .

then $(G,\rho)$ is called the group.

Additionally, if $a\rho b=b\rho a$ (commutativity) is satisfied at all times, the groups is called the abelian group.

If the operation $\rho$ may be $+$ , $(\mathbb{Z},+)$ is one of abelian groups,
and the element $e$ is $0$ ,and the element $y$ is $-a$.

When the set $\mathbb{R}^*$means real numbers except $0$ , $(\mathbb{R}^*, \times)$ becomes the abelian group.
(you must know $e=1$ and $y=\frac{1}{a}$.)

The collection of N-square matrix becomes a group (not an abelian group).

A group is an algebraic structure.