Supersolvable Group

Definition

Let $G$ be a group. $G$ is supersolvable if there exists a normal series

$$\{1\}=H_0◃H_1◃\cdots ◃H_{s-1}◃H_s=G$$

such that each quotient group $H_{i+1}/H_i$ is cyclic and each $H_i$ is normal in $G$.

Remark.

• solvable group: Each quotient is abelian.
• polycyclic group: no requirement that each $H_i$ be normal in $G$.
• Each finite solvable group is polycyclic.
• $A_4$ is solvable but not supersolvable.