nLab
order of a group

Contents

This entry is about the concept in group theory. See at order for the concept in order theory.


Contents

Definition

Order of a group

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

Hence for GG a finite group, its order |G|\vert G\vert \in \mathbb{N} is a natural number, the number of its group elements.

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

The exponent of a group is the least common multiple of the order of all elements of the group.

Order 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.

Properties

For GG a finite group and gGg \in G an element, there is a close relation between

  1. the order |G|\vert G\vert of GG in the sense above;

  2. the order order(g)order(g) of gg the sense above

Orders of cyclic subgroups

The element gg generates a cyclic subgroup C gGC_g \subset G. Evidently, the order of the element gg equals the order |C g|\vert C_g \vert of this cyclic subgroup that it generates:

(1)order(g)=|C g|. order(g) =\vert C_g\vert \,.

Lagrange’s theorem

By Lagrange's theorem, if GG a finite group and HGH \subset G a subgroup, then the order |G|\vert G\vert of GG is divisible by the order |H|\vert H\vert of HH. The multiple is called the subgroup index [G:H][G : H] \in \mathbb{N}:

|G|=[G:H]|H|. {\vert G\vert} \;=\; [G : H] \, {\vert H\vert} \,.

Hence with (1) it follows that with gGg \in G any element, the order of gg divides the order of GG:

|G|/order(g)=[G:C g]. {\vert G\vert}/ order(g) \;=\; [G : C_g] \,.

References

See also

Last revised on October 5, 2018 at 02:37:05. See the history of this page for a list of all contributions to it.