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:

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

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 $p^{v h}$; in this case $h$ is called the order or height of $X$.

Revised on October 17, 2012 13:40:57
by Urs Schreiber
(82.169.65.155)