Supersolvable group

Remarks

Example: $S_3$ is supersolvable with the series $\{e\} \le A_3 \le S_3$. Here $A_3$ and $S_3$ are normal in $S_3$, $A_3 \cong \Z/3\Z$ and $S_3/A_3 \cong \Z/2\Z$. Non-example: $A_4$ is not supersolvable. Its only normal subgroups are $\{e\}$, the Klein four-group $V$ and $A_4$. So the first term of such a series that differs from $\{e\}$ is $V$ or $A_4$, and the factor at that step is $V$ or $A_4$, which is not cyclic. Every finitely generated nilpotent group is supersolvable, and every supersolvable group is polycyclic, hence solvable. Some authors write supersoluble.