Homomorphism theorem and group product

Question

I started studying group theory and I am puzzled about a manipulation of the homomorphism theorem.

It says:

$$G/{\ker}(\phi) \simeq {\rm im}(\phi)$$

but can it also imply:

$$ G = {\rm ker}(\phi) \times {\rm im}(\phi)$$

Is it always true for a finite group G and in which case is it true for infinite group G (if classifiable)?

Answer 1

Suppose that we have dihedral group $D_3$ and $\phi=sgn$, then ${\rm im} \phi=\mathbb{Z}_2$, $\ker \phi=\mathbb{Z}_3$ but

$$D_3 \neq \mathbb{Z}_3 \times \mathbb{Z}_2.$$