# Axiom:Finite Group/Axioms

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## Definition

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 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): Chapter $\text {I}$: The Group Concept: $\S 4$: Alternative Axioms for Finite Groups: Theorem $1$