Inverse in Group is Unique
Theorem
Let $\struct {G, \circ}$ be a group.
Then every element $x \in G$ has exactly one inverse:
- $\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x^{-1} \circ x$
where $e$ is the identity element of $\struct {G, \circ}$.
Proof 1
By the definition of a group, $\struct {G, \circ}$ is a monoid each of whose elements has an inverse.
The result follows directly from Inverse in Monoid is Unique.
$\blacksquare$
Proof 2
Let $\struct {G, \circ}$ be a group whose identity element is $e$.
By Group Axiom $\text G 3$: Existence of Inverse Element, every element of $G$ has at least one inverse.
Suppose that:
- $\exists b, c \in G: a \circ b = e, a \circ c = e$
that is, that $b$ and $c$ are both inverse elements of $a$.
Then:
\(\ds b\) | \(=\) | \(\ds b \circ e\) | as $e$ is the identity element | |||||||||||
\(\ds \) | \(=\) | \(\ds b \circ \paren {a \circ c}\) | as $c$ is an inverse of $a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {b \circ a} \circ c\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds e \circ c\) | as $b$ is an inverse of $a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds c\) | as $e$ is the identity element |
So $b = c$ and hence the result.
$\blacksquare$
Proof 3
Let $x, y \in G$.
We already have, from the definition of inverse element, that:
- $\forall x \in G: \exists x^{-1} \in G: x \circ x^{-1} = e = x^{-1} \circ x$
By Group has Latin Square Property, there exists exactly one $a \in G$ such that $a \circ x = y$.
Similarly, there exists exactly one $b \in G$ such that $x \circ b = y$.
Substituting $e$ for $y$, it follows that $x^{-1}$ as defined above is unique.
$\blacksquare$
Also see
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$: Exercise $5$