Induced Group Product is Homomorphism iff Commutative/Corollary
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Corollary to Induced Group Product is Homomorphism iff Commutative
Let $\struct {G, \circ}$ be a group.
Let $\phi: G \times G \to G$ be defined such that:
- $\forall a, b \in G: \map \phi {a, b} = a \circ b$
Then $\phi$ is a homomorphism if and only if $G$ is abelian.
Proof
We have that $G$ is a subgroup of itself.
The result then follows from Induced Group Product is Homomorphism iff Commutative by putting $H_1 = H_2 = G$.
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 60 \epsilon$