The Identity Element in Groups

In any given group (G,⋆), there is a remarkable element known as the "identity element." It has a distinctive property: when it's involved in any operation with another element of the group, the latter remains unaffected.

Take multiplication of numbers as an example: the identity element is 1. Multiplying any number by 1 leaves it unchanged.

On the other hand, for addition, the identity element is 0. Adding 0 to any number does not alter its value in the slightest.

Illustrating with an Example

Let's take a concrete example, such as the set of all integers (denoted as Z) with addition (+) as our operation of focus.

In this context (Z,+), 0 serves as the identity element.

When you add 0 to any number, there's no impact on the original number. Whether it's 1, which stays 1, or -2, which remains -2.

$$ 1 + 0 = 1 $$

$$ -2 + 0 = -2 $$

This concept holds true across the board.

A Unique Identity Element for Every Group

This aspect is far from trivial because it indicates that whenever you identify an element fulfilling this unique role, it's the sole element that can do so within that specific framework.