Definition:Group Axioms

From ProofWiki

Jump to: navigation, search

Definition

A group is an algebraic structure $\left({G, \circ}\right)$ which satisfies the following four conditions:

$(G0):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall a, b \in G$	$:$	$\displaystyle a \circ b \in G$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Closure Axiom
$(G1):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall a, b, c \in G$	$:$	$\displaystyle a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ c$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Associativity Axiom
$(G2):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \exists e \in G: \forall a \in G$	$:$	$\displaystyle e \circ a = a = a \circ e$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Identity Axiom	The element $e$ is called the identity of $G$.
$(G3):$	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \forall a \in G: \exists b \in G$	$:$	$\displaystyle a \circ b = e = b \circ a$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Inverse Axiom	The element $b$ is called the inverse of $a$ and is usually denoted $a^{-1}$.

These four stipulations are called the group axioms.

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

\((G0):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall a, b \in G\)	\(:\)	\(\displaystyle a \circ b \in G\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Closure Axiom
\((G1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall a, b, c \in G\)	\(:\)	\(\displaystyle a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity Axiom
\((G2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \exists e \in G: \forall a \in G\)	\(:\)	\(\displaystyle e \circ a = a = a \circ e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Identity Axiom	The element $e$ is called the identity of $G$.
\((G3):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall a \in G: \exists b \in G\)	\(:\)	\(\displaystyle a \circ b = e = b \circ a\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Inverse Axiom	The element $b$ is called the inverse of $a$ and is usually denoted $a^{-1}$.