Product Inverse Operation Properties induce Group

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {G, \oplus}$ be a closed algebraic structure on which the following properties hold:

\((\text {PI} 1)\)   $:$   Self-Inverse Property      \(\ds \forall x \in G:\) \(\ds x \oplus x = e \)      
\((\text {PI} 2)\)   $:$   Right Identity      \(\ds \exists e \in G: \forall x \in G:\) \(\ds x \oplus e = x \)      
\((\text {PI} 3)\)   $:$   Product Inverse with Right Identity      \(\ds \forall x, y \in G:\) \(\ds e \oplus \paren {x \oplus y} = y \oplus x \)      
\((\text {PI} 4)\)   $:$   Cancellation Property      \(\ds \forall x, y, z \in G:\) \(\ds \paren {x \oplus z} \oplus \paren {y \oplus z} = x \oplus y \)      

These four stipulations are known as the product inverse operation axioms.


Let $\circ$ be the operation on $G$ defined as:

$\forall x, y \in G: x \circ y = x \oplus \paren {e \oplus y}$


Then $\struct {G, \circ}$ is a group.


Proof

Taking the group axioms in turn:


Group Axiom $\text G 0$: Closure

$\struct {G, \oplus}$ is closed by definition.

Hence $\struct {G, \circ}$ is likewise closed.

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element

\(\ds x \circ e\) \(=\) \(\ds x \oplus \paren {e \oplus e}\) Definition of $\circ$
\(\ds \) \(=\) \(\ds x \oplus e\) $\text {PI} 1$: Self-Inverse Property
\(\ds \) \(=\) \(\ds x\) $\text {PI} 2$: Right Identity

and:

\(\ds e \circ x\) \(=\) \(\ds e \oplus \paren {e \oplus x}\) Definition of $\circ$
\(\ds \) \(=\) \(\ds x \oplus e\) $\text {PI} 3$: Product Inverse with Right Identity
\(\ds \) \(=\) \(\ds x\) $\text {PI} 2$: Right Identity


Thus $e$ is the identity element of $\struct {G, \circ}$.

$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element

We have that $e$ is the identity element of $\struct {G, \circ}$.

Hence:

\(\ds \forall x \in G: \, \) \(\ds x \circ \paren {e \oplus x}\) \(=\) \(\ds x \oplus \paren {e \oplus \paren {e \oplus x} }\) Definition of $\circ$
\(\ds \) \(=\) \(\ds x \oplus \paren {x \oplus e}\) $\text {PI} 3$: Product Inverse with Right Identity
\(\ds \) \(=\) \(\ds x \oplus x\) $\text {PI} 2$: Right Identity
\(\ds \) \(=\) \(\ds e\) $\text {PI} 1$: Self-Inverse Property


and:

\(\ds \forall x \in G: \, \) \(\ds \paren {e \oplus x} \circ x\) \(=\) \(\ds \paren {e \oplus x} \oplus \paren {e \oplus x}\) Definition of $\circ$
\(\ds \) \(=\) \(\ds e \oplus e\) $\text {PI} 4$: Cancellation Property
\(\ds \) \(=\) \(\ds e\) $\text {PI} 1$: Self-Inverse Property


Thus every element $x$ of $\struct {G, \circ}$ has inverse $e \oplus x$.

$\Box$


Group Axiom $\text G 1$: Associativity

We have:

\(\ds \forall x, y, z \in G: \, \) \(\ds x \circ \paren {y \circ z}\) \(=\) \(\ds x \circ \paren {y \oplus \paren {e \oplus z} }\) Definition of $\circ$
\(\ds \) \(=\) \(\ds \paren {\paren {x \circ y} \oplus y} \circ \paren {y \oplus \paren {e \oplus z} }\) Product Inverse Operation Properties: Lemma $5$: substituting $\paren {x \circ y} \oplus y$ for $x$
\(\ds \) \(=\) \(\ds \paren {x \circ y} \oplus \paren {e \oplus z}\) Product Inverse Operation Properties: Lemma $2$
\(\ds \) \(=\) \(\ds \paren {x \circ y} \circ z\) Definition of $\circ$

Thus $\circ$ is associative.

$\Box$


All the group axioms are thus seen to be fulfilled, and so $\struct {G, \circ}$ is a group.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Product_Inverse_Operation_Properties_induce_Group&oldid=544625"