Finite Group Order

The order of a finite group refers to its total number of elements, or its cardinality. For a finite group \( G \), the order, denoted \( |G| \), is a positive integer that reflects the size of the group.

Consider a group \( G \) comprised of the elements \(\{e, a, b, c\}\), with \( e \) serving as the identity element. The order of \( G \) is then 4, indicating the presence of four distinct elements within the group.

Conversely, the order of an individual element in a finite group is identified as the smallest positive integer \( n \) for which \( g^n = e \), where \( g \) represents an element of the group and \( e \) is the identity element.

If no such \( n \) can be found, the element is considered to have an infinite order.

Yet, within a finite group, each element's order must divide the group's overall order, as per Lagrange's theorem. This ensures that every element within a finite