Identity is Only Group Element of Order 1

Theorem

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


Proof

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

Then:

$e^1 = e$

and:

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

Hence the result.

$\blacksquare$


Sources

  • 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): $\S 8$: Chapter $\text {I}$: The Group Concept: The Order (Period) of an Element: $\text{(i)}$
  • 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.4$. Cyclic groups: Example $101$
  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 38.1$ Period of an element
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