# Definition:Finite Group/Axioms

## Contents

## Definition

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 known as

The **finite group axioms** are also known as the **finite group postulates**, but the latter term is less indicative of the nature of these statements.

The numbering of the axioms themselves is to a certain extent arbitrary.

## Also see

## Sources

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): $\S 4$: Alternative Axioms for Finite Groups: Theorem $1$