Identity of Group is Unique

Theorem

Let $\left({G, \circ}\right)$ be a group which has an identity element $e \in S$.

Then $e$ is unique.

Proof

By the definition of a group, $\left({G, \circ}\right)$ is also a semigroup with an identity element.

The result follows by applying the result Identity of Semigroup is Unique.

$\blacksquare$

Sources

• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.4$: Lemma $2$
• 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 $3.1$