If $K$ is a normal subgroup of $G$, is it true that $G$ is isomorphic to the direct product $K \times (G / K)$?

Question

Let $K$ be a normal subgroup of the group $G$. Is it true that the direct product of $K$ by $G/K$ is isomorphic to $G$? Which isomorphism can we define from $K\times G/K$ to $G$?

Answer 1

In general this is false. A minimal example is $G := \Bbb Z_4$, $K := \Bbb Z_2$. Then, $G / K \cong \Bbb Z_2$, but then $G$ has elements of order $4$ whereas $K \times (G / K)$ does not, so $$G \not\cong K \times (G / K).$$

Another easy example is $G := S_3$, $K := A_3 \cong \Bbb Z_3$, for which $G$ is nonabelian but $K \times (G / K)$ is not.