order of an element

The order of an element measures how long it takes to “come full circle.” Keep applying the same element to itself; the order is the number of steps before you return to the identity for the first time. On a clock, stepping forward 4 hours at a time, you return to start after 3 steps (4, 8, 12) — so that step has order 3.

Formally, in a group with identity e, the order of an element a is the smallest positive integer n such that a^n = e (using additive notation, the smallest n with n·a = e). If no such n exists — you can keep applying a forever without returning to e — the element has infinite order.

Order is a fundamental fingerprint of an element. The element generates a cyclic subgroup whose size equals its order, and by Lagrange's theorem that size must divide the order of a finite group. Don't confuse the order of an element (a property of one element) with the order of the group (the number of elements it contains).