Induced Group Product is Homomorphism iff Commutative

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Theorem

Let $\struct {G, \circ}$ be a group.

Let $H_1, H_2$ be subgroups of $G$.

Let $\phi: H_1 \times H_2 \to G$ be defined such that:

$\forall \tuple {h_1, h_2} \in H_1 \times H_2: \map \phi {h_1, h_2} = h_1 \circ h_2$


Then $\phi$ is a homomorphism if and only if every element of $H_1$ commutes with every element of $H_2$.


Corollary

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 $\tuple {h_1, h_2} \circ \tuple {k_1, k_2} = \tuple {h_1 \circ k_1, h_2 \circ k_2}$ by definition of group direct product.


Necessary Condition

Let $\phi$ be a homomorphism.

Then:

\(\displaystyle \map \phi {\tuple {h_1, h_2} \circ \tuple {k_1, k_2} }\) \(=\) \(\displaystyle \map \phi {h_1, h_2} \circ \map \phi {k_1, k_2}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \phi {h_1 \circ k_1, h_2 \circ k_2}\) \(=\) \(\displaystyle \tuple {h_1 \circ h_2} \circ \tuple {k_1 \circ k_2}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle h_1 \circ k_1 \circ h_2 \circ k_2\) \(=\) \(\displaystyle h_1 \circ h_2 \circ k_1 \circ k_2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle k_1 \circ h_2\) \(=\) \(\displaystyle h_2 \circ k_1\)

This follows whatever $k_1$ and $h_2$ are.

So in order for $\phi$ to be a homomorphism, every element of $H_1$ must commute with every element of $H_2$.

$\Box$


Sufficient Condition

Let every element of $H_1$ commute with every element of $H_2$.

Let $\tuple {h_1, h_2}, \tuple {k_1, k_2} \in H_1 \times H_2$.


Then:

\(\displaystyle \map \phi {\tuple {h_1, h_2} \circ \tuple {k_1, k_2} }\) \(=\) \(\displaystyle \map \phi {h_1 \circ k_1, h_2 \circ k_2}\)
\(\displaystyle \) \(=\) \(\displaystyle h_1 \circ k_1 \circ h_2 \circ k_2\)
\(\displaystyle \) \(=\) \(\displaystyle h_1 \circ h_2 \circ k_1 \circ k_2\) $k_1$ commutes with $h_2$
\(\displaystyle \) \(=\) \(\displaystyle \map \phi {h_1, h_2} \circ \map \phi {k_1, k_2}\)

Thus $\phi$ is shown to be a homomorphism.

$\blacksquare$