# Theorem In [[group|groups]] of finite [[Group Order|order]], if an [[Group Element|element]] $g$ is multiplied by itself enough times, say $n$, we recover the identity, i.e. $g^n = e$. ### Proof If the [[group]] is finite, multiplying an element by itself a sufficient number of times will always give rise to a repetition. Let $h$ represent the repetition: $ h = g^p = g^q, \quad \text{where } p > q $ Let $p = q + n$, so we have: $ h = g^p = g^{q+n} = g^q g^n = h = g^q $ Thus, we have $ g^n = e $ # Definition The order of a group element $g$ is the smallest value of $n$ that satisfies the relationship $g^n= E$. It is denoted by $ord(g)$ or $|g|$. The identity element of any group is obviously always of order 1. If no such $n$ exists, then the element is said to have infinite order. The order of an element is equal to the order of its [[Cyclic Subgroup]] $\langle g\rangle$