Mapping to Square is Endomorphism iff Abelian
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $\phi: G \to G$ be defined as:
- $\forall g \in G: \map \phi g = g \circ g$
Then $\struct {G, \circ}$ is abelian if and only if $\phi$ is a (group) endomorphism.
Proof
Necessary Condition
Let $\struct {G, \circ}$ be an abelian group.
Let $a, b \in G$ be arbitrary.
Then:
\(\ds \map \phi {a \circ b}\) | \(=\) | \(\ds \paren {a \circ b} \circ \paren {a \circ b}\) | Definition of $\phi$ | |||||||||||
\(\ds \) | \(=\) | \(\ds a \circ \paren {b \circ a} \circ b\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds a \circ \paren {a \circ b} \circ b\) | Definition of Abelian Group: Commutativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ a} \circ \paren {b \circ b}\) | Group Axiom $\text G 1$: Associativity | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \phi a \circ \map \phi b\) | Definition of $\phi$ |
As $a$ and $b$ are arbitrary, the above holds for all $a, b \in G$.
Thus $\phi$ is a group homomorphism from $G$ to $G$.
So by definition, $\phi$ is a group endomorphism.
$\Box$
Sufficient Condition
Let $\phi: G \to G$ as defined above be a group endomorphism.
Then:
\(\ds \forall a, b \in G: \, \) | \(\ds \map \phi {a \circ b}\) | \(=\) | \(\ds \map \phi a \circ \map \phi b\) | Definition of Group Endomorphism | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a, b \in G: \, \) | \(\ds \paren {a \circ b} \circ \paren {a \circ b}\) | \(=\) | \(\ds \paren {a \circ a} \circ \paren {b \circ b}\) | Definition of $\phi$ | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a, b \in G: \, \) | \(\ds a \circ \paren {b \circ a} \circ b\) | \(=\) | \(\ds a \circ \paren {a \circ b} \circ b\) | Group Axiom $\text G 1$: Associativity | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a, b \in G: \, \) | \(\ds b \circ a\) | \(=\) | \(\ds a \circ b\) | Cancellation Laws |
Thus, by definition, $G$ is an abelian group.
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 60 \delta$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Exercise $(9)$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Exercise $6$