# Definition:Finite Group

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## 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 $\struct {G, \circ}$ which satisfies the following four conditions:

\((\text {FG} 0)\) | $:$ | Closure | \(\displaystyle \forall a, b \in G:\) | \(\displaystyle a \circ b \in G \) | ||||

\((\text {FG} 1)\) | $:$ | Associativity | \(\displaystyle \forall a, b, c \in G:\) | \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \) | ||||

\((\text {FG} 2)\) | $:$ | Finiteness | \(\displaystyle \exists n \in \N:\) | \(\displaystyle \order G = n \) | ||||

\((\text {FG} 3)\) | $:$ | 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*.

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 4$: Alternative Axioms for Finite Groups - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.4$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 38$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(6)$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Definition $1.2$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**order**:**6.**