Definition:Finite Group

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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 $\struct {G, \circ}$ 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.

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.