> [!NOTE] **Definition** (Order of group element) > The *order* of a [[Groups|group]] element $a$ is the smallest $n \in \mathbb{N}^{+}$ such that its [[Integer Power of Group Element|power]] satisfies $a^{n} = 1$If there is no such positive integer $n$, we say $a$ has *infinite order*. > # Properties By [[Identity is the only group element of order 1]], the identity element is the only group element of order $1.$ By [[Power of Group Element is Identity only If Order divides]], $a^{m}=1$ iff $n\mid m$ where $n$ is the order of the group element $a.$ # Applications **Finite groups**: Every member of [[Finite Group|finite group]] has finite order. A consequence of *Lagrange's theorem* is that [[Order of Element of Finite Group Divides Order of The Group|the order of each member of a finite group divides the order of the group]].