How does an identity element work in a mathematical group?

In modern algebra, an identity element is a fundamental component of a group. It functions as a neutral element within the set, meaning that when it is combined with any other element of the group using the group's binary operation, the original element remains unchanged.12

Formally, if $e$ represents the identity element and $a$ is any element in the group, the identity property is defined by the equation:
$$a*e=a=e*a$$

This property ensures that the identity element acts as a "do-nothing" operation within the system. For example, in the set of integers under addition, the identity element is $0$, because adding $0$ to any integer does not change its value. Similarly, in the set of positive rational numbers under multiplication, the identity element is $1$, as multiplying any number by $1$ leaves it unchanged. The existence of this element is one of the essential axioms required for a mathematical system to be classified as a group.312