# Order of a group

## Contents

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## Definition

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 $\left| G \right|$.

## Facts

### Subgroup

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.

### Quotient

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

## Computation

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

Order (group);
where
group[/itex] may either be an on-the-spot definition of a group or a name for something defined earlier.