nLab
order of a group

Contents

Definition

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

Other meanings

Order of an element of a group

For GG a group and gGg \in G an element, the order of gg is the smallest natural number nn such that the nn-fold group product of gg with itself is the neutral element:

order(g)min{ng 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 XX over a field (of characteristic pp) which is defined to be the dimension of the associated ring of functions O(X)O(X) as a kk-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 vhp^{v h}; in this case hh is called the order or height of XX.

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