nLab order of a group

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

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Definition

Order of a group

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

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

Order of an element of a group

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

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

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 $X$ 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 $X$.

Properties

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

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

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

Orders of cyclic subgroups

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

(1)$$order(g) =\vert C_g\vert \,.$$

Lagrange’s theorem

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

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

Hence with (1) it follows that with $g \in G$ any element, the order of $g$ divides the order of $G$:

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

Related concepts

• exponent of a group

• p-group

References

See also

• Wikipedia, Order (group theory)