Product Inverse Operation Properties induce Group
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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.7 \ \text {(b)}$