Definition of Group

A group is a set $G$ equipped with a binary operation $\ast$ that satisfies the following four properties:

Closure:
For any two elements $a,b\in G$, $a\ast b$ is also in $G$.

Associativity:
For any three elements $a,b,c\in G$, $(a\ast b)\ast c=a\ast (b\ast c)$.

Identity Element:
There exists an element $e\in G$ such that for every element $a\in G$, $a\ast e=e\ast a=a$.

Inverse Element:
For every element $a\in G$, there exists an element $b\in G$ such that $a\ast b=b\ast a=e$, where $e$ is the identity element.

Examples: non-zero real numbers

Let $G={\mathbb{R}}^{\ast }$ (the set of all non-zero real numbers) and define the operation $\cdot$ as multiplication.

Closure:
The product of any two non-zero real numbers is also a non-zero real number.

Associativity:
Multiplication is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.

Identity Element:
The identity element is $1$, because for any non-zero real number $a$, $a\cdot 1=1\cdot a=a$.

Inverse Element:
For each non-zero real number $a$, its inverse is $1/a$, because $a\cdot (1/a)=(1/a)\cdot a=1$.

Thus, $({\mathbb{R}}^{\ast },\cdot )$ is a group.