Identity Element is Idempotent
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $e \in S$ be an identity with respect to $\circ$.
Then $e$ is idempotent under $\circ$.
Proof
By the definition of an identity element:
- $\forall x \in S: e \circ x = x$
Thus in particular:
- $e \circ e = e$
Therefore $e$ is idempotent under $\circ$.
$\blacksquare$
Also see
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory: $(1.8)$