4.57 (1973)

Let a group $G$ be the product of two of its abelian minimax subgroups $A$ and $B$. Prove or refute the following statements:
$\qquad$ a) $A_0 B_0 \neq 1$, where $A_0 = \bigcap_{x \in G} A^x$ and similarly for $B_0$;
$\qquad$ b) the derived subgroup of $G$ is a minimax subgroup.