Inverse in Group is Unique/Proof 1

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

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.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 33.1$. The definition of a group