Sylow tower

Definition

A Sylow tower in a finite group $G$ is a normal series $1=P_{0}\leq P_{1}\leq \dots \leq P_{r}=G$ satisfying the following equivalent conditions:

• For every $p$ dividing the order of the group, there is a unique quotient $P_{k}/P_{k-1}$ of successive groups in the tower that has order a power of $p$. All quotients of successive groups in the tower have prime power orders.
• For every $p$ dividing the order of the group, there is a unique quotient $P_{k}/P_{k-1}$ isomorphic to a Sylow $p$-subgroup of $G$. Further, every quotient $P_{k}/P_{k-1}$ is isomorphic to some Sylow $p$-subgroup of $G$.

A finite group need not have a Sylow tower. A finite group that does have a Sylow tower is termed a group having a Sylow tower.