Identity is only Idempotent Element in Group/Proof 2
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Theorem
Every group has exactly one idempotent element: the identity.
Proof
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Let $x \in G$ such that $x \circ x = x$.
\(\ds e\) | \(=\) | \(\ds x \circ x^{-1}\) | Group Axiom $\text G 3$: Existence of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x \circ x} \circ x^{-1}\) | by hypothesis: $x \circ x = x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x \circ \paren {x \circ x^{-1} }\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds x \circ e\) | Definition of Inverse Element | |||||||||||
\(\ds \) | \(=\) | \(\ds x\) | Definition of Identity Element |
$\blacksquare$
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: $(1.14)$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.4$: Lemma $1$
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$ Semigroups, Monoids and Groups: Theorem $1.2 \text{(i)}$