# Tour:Finite group

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Mathematics doesn't hesitate to study the infinite. But the structure and nature of finite groups is, in general, very different from that of infinite groups.
WHAT YOU NEED TO DO: Read quickly the definition below for a finite group; this concept will play an important role throughout group theory.

## Definition

A group $G$ is said to be finite if the cardinality of its underlying set (i.e., its order) is finite. Here, the cardinality of a set refers to the number of elements in the set, and is denoted as $|G|$.

## Examples

The trivial group is an example of a finite group -- the underlying set has cardinality one. Other examples of finite groups include the symmetric group on a set, and the cyclic group of order $n$. Any subgroup of a finite group is finite.

The group of integers, group of rational numbers, and group of real numbers (each under addition) are not finite groups.