Order of a group

From Groupprops
Revision as of 15:45, 1 July 2008 by Vipul (talk | contribs) (Definition)
Jump to: navigation, search
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
VIEW: Definitions built on this | Facts about this: (facts closely related to Order of a group, all facts related to Order of a group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |[SHOW MORE]
This article defines an arithmetic function on groups
View other such arithmetic functions


QUICK PHRASES: size of a group, cardinality of a group, size of the underlying set, number of elements

Symbol-free definition

The order of a group is the cardinality of its underlying set.

Definition with symbols

The order of a group G is the cardinality of G as a set. it is denoted as \left| G \right|.



By Lagrange's theorem, the order of any subgroup divides the order of the group.

The converse is not always true, that is, there may exist numbers dividing the order of the group with no subgroups of those orders.

In particular, this also means that the order of an element in the group divides the order of the group. Hence, the exponent of a group divides its order.


The order of any quotient of a group also divides the order of the group.


Template:GAP command for function

The GAP command to compute the order of a group is:

Order (group);
group</math> may either be an on-the-spot definition of a group or a name for something defined earlier.


Textbook references