# Tour:Order of a group

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WHAT YOU NEED TO DO: Go through the definition below and understand it.

## 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 (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.