# Definition: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 $\displaystyle \forall a, b \in G:$ $\displaystyle a \circ b \in G$ $(\text {FG} 1)$ $:$ 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 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.