Order of a group

Symbol-free definition
The order of a group is the cardinality (i.e., size, or number of elements) of its underlying set.

Definition with symbols
The order of a group $$G$$ is the cardinality (i.e., size, or number of elements) of $$G$$ as a set. it is denoted as $$\left| G \right|$$.

Note that a finite group is a group whose underlying set is finite, i.e., the size of the underlying set is finite. The order of a finite group is thus a natural number (note that the order cannot be zero because every group contains the identity element and is hence nonempty). For an infinite group, the order is an infinite cardinal.

Examples

 * The trivial group, which is the group with only the identity element, has order $$1$$. In fact, it is the only group (up to isomorphism}) that has order 1.

(see the next section for more examples).

Operations that do not admit a clear formula
These include, for instance, measuring the sizes of homomorphism sets between two groups.

Particular cases and general patterns
See also number of groups of given order for more information on how the number of groups of order $$n$$ depends on $$n$$.

Divisibility relations for other arithmetic functions
Any arithmetic function that arises as the maximum or lcm of a bunch of numbers each of which divides the order of the group, must again divide the order of the group. Some examples are given below:

Analogues
An order is a size measure that works for finite groups. For infinite groups, the order, viewed as an infinite cardinal, is a very crude size measure since it is unable to differentiate between the group and subgroups of finite index. Some analogues that work are described below.


 * Measure is an analogue that is used for amenable groups, and works particularly well for compact groups with a Haar measure.
 * Dimension plays a role analogous to the logarithm of the order. In those situations where the orders multiply, dimensions tend to add. There are many different notions of dimension, including algebraic, analytic, and topological ones.
 * For profinite groups, we can view the orders as supernatural numbers, i.e., the orders take values that are products of powers of possibly infinitely many primes. Analogues of Lagrange's and Sylow's theorems hold in these contexts.

Order from subgroup and quotient perspectives
An infinite cyclic group (i.e., a group isomorphic to the group of integers) has no proper nontrivial finite subgroups, so from the perspective of Lagrange's theorem for subgroups, its order can be thought of as having no prime divisors. However, it has finite quotients of every order, so from the perspective of the fact that order of quotient group divides order of group, every prime divides its order.

More generally, for periodic groups and locally finite groups, the order notion must capture the possible finite subgroups that arise, whereas for residually finite groups, the order notion must capture the possible finite quotients that arise.

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

Order (group);

where group may either be an on-the-spot definition of a group or a name for something defined earlier.

Textbook references

 * , Page 47, Point (2.10)