# Tour:Finite group

From Groupprops

**This article adapts material from the main article:** finite group

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)PREVIOUS: Order of a group|UP: Introduction two (beginners)|NEXT: Subsemigroup of finite group is subgroupExpected time for this page: 2 minutes

General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part

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 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 .

## 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 . 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.

This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.PREVIOUS: Order of a group|UP: Introduction two (beginners)|NEXT: Subsemigroup of finite group is subgroup