Does an isomorphism of groups that can be written as a direct product induce isomorphisms on the factors?

Solution 1:

No, and this is always false for any group that can be written as a direct product in a non-trivial way. For example, if $G = A \times B$ with neither $A$ nor $B$ the trivial group, then

$$A \times B = G \equiv G \times \{e\}$$

Solution 2:

Yet another example: let $A\not\cong B$. Then $A\times B\cong B\times A$ . . .

Solution 3:

I would add to the answers already provided that even if $A_1\cong X_1$ and $A_2\cong X_2$, there may not exist isomorphisms $\phi_1:A_1\to X_1$ and $\phi_2:A_2\to X_2$ such that $\phi(a_1,a_2)=(\phi_1(a_1),\phi_2(a_2))$. For instance, let $A_1=A_2=X_1=X_2=\mathbb{Z}$ and consider $\phi:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}$ given by $\phi(a,b)=(a,b+a)$. Then $\phi$ is an isomorphism (its inverse is given by $(a,b)\mapsto(a,b-a)$), but it cannot come from a pair of isomorphisms $\phi_1$ and $\phi_2$ because the second coordinate of $\phi$ depends on both coordinates of the input.