Identity Only Group Element Order 1

Theorem

In every group, the identity, and only the identity, has order $1$.

Proof

Let $G$ be a group with identity $e$.

• $e^1 = e$.

• $\forall a \in G: a \ne e: a^1 = a \ne e$.

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.4$: Example $101$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 38.1$