nLab
order of a group

Definition

For $G$ a discrete group, its order is the cardinality of the underlying set.

Other meanings

Order of an element of a group

For $G$ a group and $g\in G$ an element, the order of $g$ is the smallest natural number $n$ such that the $n$-fold group product of $g$ with itself is the neutral element:

$$\mathrm{order}(g)≔\min \{n\in \mathbb{N}\mid g^n=e\}.$$order(g) \coloneqq min \{ n \in \mathbb{N} | g^n = e\} \,.

Of a group scheme

Sometimes the term ”order” refers to the height of a (group) scheme $X$ over a field (of characteristic $p$) which is defined to be the dimension of the associated ring of functions $O(X)$ as a $k$-vector space. Another term for this notion is ”rank”. If this group scheme is moreover p-divisible - which means that is is in fact a codirected diagram of group schemes of order $p^{vh}$; in this case $h$ is called the order or height of $X$.