The Identity Element in a Group

Within a group (G,*), the element e in G is referred to as the "identity element" (or "neutral element") with respect to the binary operation *. This means for every element g in G, the equation $$ g*e=e*g=g $$ always holds.

The identity element is unique in that it, when combined with any other element in the group through the group's operation, does not alter that element.

Simply put, engaging any group element with the identity element yields that same element unchanged.

For instance, in a multiplicative group, the identity is often represented as 1 or e. Conversely, in an additive group, the identity is denoted by 0. It's crucial to recognize that a group's identity element can be symbolized in various ways, provided its representation is clear.

Practical Example

Take the set of integers Z, combined with the operation of addition (+).

Here, the identity element is 0.

This is because, for any integer a within Z, the following is true: a+0=0+a=a.

$$ a+0=0+a=a $$

Therefore, adding zero to any integer invariably returns the original integer.

Examples include:

$$ 3+0=0+3=3 $$

$$ (-5)+0=0+(-5)=-5 $$

In this context, the identity element is 0, indicating that adding zero to any integer maintains its value unchanged.

Insights

Observations on the identity element within a group include:

• Uniqueness of the Identity Element
  A group (G,*) has precisely one identity element with respect to the operation *.

  Demonstration. Suppose, for the sake of contradiction, there were two identity elements "e" and "u". Then, for every element g in G, the following must hold: $$ \forall \ g \in G \ \ eg=ge=g $$ and $$ \forall \ g \in G \ \ ug=gu=g $$. Now, applying identity element u to e (and vice versa) based on the definition of an identity element, results in: $$ ue=eu=e $$ and likewise $$ eu=ue=u $$, demonstrating that u and e must be the same element as they produce the same outcome in both operations (ue=eu=e=u).

And so forth.