 Order (group theory)

In group theory, order can mean the cardinality of a group or the least positive $n$ for which a group element $g$ satisfies $g^n=e$. If no such $n$ exists, the element has infinite order. For a finite group, Lagrange's theorem implies that each element order divides the group order.

In the context of group theory, the term "order" can refer to two related but distinct concepts: 1. **Order of a Group**: The order of a group \( G \), denoted as \( |G| \), is defined as the number of elements in the group. For finite groups, this is simply a count of all the elements.