Definition:Finite Group

Definition

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.

A group which is not finite is an infinite group.

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

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

$(\text {FG} 0)$	$:$	Closure	$\ds \forall a, b \in G:$	$\ds a \circ b \in G $
$(\text {FG} 1)$	$:$	Associativity	$\ds \forall a, b, c \in G:$	$\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c $
$(\text {FG} 2)$	$:$	Finiteness	$\ds \exists n \in \N:$	$\ds \order G = n $
$(\text {FG} 3)$	$:$	Cancellability	$\ds \forall a, b, c \in G:$	$\ds c \circ a = c \circ b \implies a = b $
				$\ds a \circ c = b \circ c \implies a = b $

These four stipulations are called the finite group axioms.

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

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

\((\text {FG} 0)\)	$:$	Closure	\(\ds \forall a, b \in G:\)	\(\ds a \circ b \in G \)
\((\text {FG} 1)\)	$:$	Associativity	\(\ds \forall a, b, c \in G:\)	\(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)
\((\text {FG} 2)\)	$:$	Finiteness	\(\ds \exists n \in \N:\)	\(\ds \order G = n \)
\((\text {FG} 3)\)	$:$	Cancellability	\(\ds \forall a, b, c \in G:\)	\(\ds c \circ a = c \circ b \implies a = b \)
				\(\ds a \circ c = b \circ c \implies a = b \)