Finite solvable groups in which the Sylow $p$-subgroups are either bicyclic or of order $p^3$

All groups considered in this paper will be finite. Our main result here is the following theorem. Let $G$ be a solvable group in which the Sylow $p$-subgroups are either bicyclic or of order $p^3$ for any $p\in\pi(G)$. Then the derived length of $G$ is at most 6. In particular, if $G$ is an $\mathrm A_4$- free group, then the following statements are true: (1) $G$ is a dispersive group; (2) if no prime $q\in\pi(G)$ divides $p^2+p+1$ for any prime $p\in\pi(G)$, then $G$ is Ore dispersive; (3) the derived length of $G$ is at most 4.