Equivalence of Axiom Schemata for Groups

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Theorem

In the definition of a group, the axioms for the existence of an identity element and for closure under taking inverses can be replaced by the following two axioms:

Given a group $G$, there exists at least one element $e \in G$ such that $e$ is a left identity;
For any element $g$ in a group $G$, there exists at least one left inverse of $g$.


Alternatively, we can also replace the aforementioned axioms with the following two:

Given a group $G$, there exists at least one element $e \in G$ such that $e$ is a right identity;
For any element $g$ in a group $G$, there exists at least one right inverse of $g$.


Thus we can formulate the group axioms as either of the following:


Group Axioms (Left)

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

\((G 0)\)   $:$   Closure Axiom      \(\displaystyle \forall a, b \in G:\) \(\displaystyle a \circ b \in G \)             
\((G 1)\)   $:$   Associativity Axiom      \(\displaystyle \forall a, b, c \in G:\) \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)             
\((G_L 2)\)   $:$   Left Identity Axiom      \(\displaystyle \exists e \in G: \forall a \in G:\) \(\displaystyle e \circ a = a \)             
\((G_L 3)\)   $:$   Left Inverse Axiom      \(\displaystyle \forall a \in G: \exists b \in G:\) \(\displaystyle b \circ a = e \)             


Group Axioms (Right)

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

\((G 0)\)   $:$   Closure Axiom      \(\displaystyle \forall a, b \in G:\) \(\displaystyle a \circ b \in G \)             
\((G 1)\)   $:$   Associativity Axiom      \(\displaystyle \forall a, b, c \in G:\) \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)             
\((G_R 2)\)   $:$   Right Identity Axiom      \(\displaystyle \exists e \in G: \forall a \in G:\) \(\displaystyle a \circ e = a \)             
\((G_R 3)\)   $:$   Right Inverse Axiom      \(\displaystyle \forall a \in G: \exists b \in G:\) \(\displaystyle a \circ b = e \)             


Proof

Suppose we define a group $G$ in the usual way, but make the first pair of axiom replacements listed above:

the existence of a left identity
every element has a left inverse.


Let $e \in G$ be a left identity and $g \in G$.

Then, from Left Inverse for All is Right Inverse, each left inverse is also a right inverse with respect to the left identity.

Also from Left Identity while exists Left Inverse for All is Identity we have that the left identity is also a right identity.

Also we have that such an Identity is Unique, so this element can rightly be called the identity.


So we have that:

$G$ has an identity;
each element of $G$ has an element that is both a left inverse and a right inverse with respect to this identity.

Therefore, the validity of the two axiom replacements is proved.

$\blacksquare$


The proof of the alternate pair of replacements (existence of a right identity and closure under taking right inverses) is similar.


Warning

Suppose we build an algebraic structure with the following axioms:

\((0)\)   $:$   Closure Axiom      \(\displaystyle \forall a, b \in G:\) \(\displaystyle a \circ b \in G \)             
\((1)\)   $:$   Associativity Axiom      \(\displaystyle \forall a, b, c \in G:\) \(\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)             
\((2)\)   $:$   Right Identity Axiom      \(\displaystyle \exists e \in G: \forall a \in G:\) \(\displaystyle a \circ e = a \)             
\((3)\)   $:$   Left Inverse Axiom      \(\displaystyle \forall x \in G: \exists b \in G:\) \(\displaystyle b \circ a = e \)             

Then this does not (necessarily) define a group (although clearly a group fulfils those axioms).


Sources