Inverses in Group are Unique

Theorem

Let $\left({G, \circ}\right)$ be a group.

Then every element $x \in G$ has exactly one inverse:

$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x \circ x$

where $e$ is the identity element of $\left({G, \circ}\right)$.

Proof

By the definition of a group, $\left({G, \circ}\right)$ is a monoid each of whose elements has an inverse.

The result follows directly from Inverses in Monoid are Unique.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.6$: Theorem $\text{(i), (ii)}$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.4$: Lemma $5$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$: Definitions $1.1 \ \text{(b)}$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 33.1$
• John F. Humphreys: A Course in Group Theory (1996): $\S 3$: Proposition $3.2$