# Definition:Finite Group

## Definition

A finite group is a group of finite order.

That is, a group $\struct {G, \circ}$ is a finite group if and only if its underlying set $G$ is finite.

That is, a finite group is a group with a finite number of elements.

### Infinite Group

A group which is not finite is an infinite group.

## Finite Group Axioms

A finite group can be defined by a different set of axioms from the conventional group axioms:

A finite group is an algebraic structure $\left({G, \circ}\right)$ which satisfies the following four conditions:

 $(FG0)$ $:$ Closure $\displaystyle \forall a, b \in G:$ $\displaystyle a \circ b \in G$ $(FG1)$ $:$ Associativity $\displaystyle \forall a, b, c \in G:$ $\displaystyle a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ c$ $(FG2)$ $:$ Finiteness $\displaystyle \exists n \in \N:$ $\displaystyle \left \lvert{G}\right \rvert = n$ $(FG3)$ $:$ Cancellability $\displaystyle \forall a, b, c \in G:$ $\displaystyle c \circ a = c \circ b \implies a = b$ $\displaystyle a \circ c = b \circ c \implies a = b$

These four stipulations are called the finite group axioms.

## Also see

• Results about the order of a group can be found here.
• Results about finite groups can be found here.

## Historical Note

The theory of finite groups was effectively originated by Augustin Louis Cauchy.

Some sources refer to him as the father of finite groups.