Induced Group Product is Homomorphism iff Commutative/Corollary

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: $\S 60 \epsilon$