Sylow tower

Definition
A Sylow tower in a finite group $$G$$ is a normal series $$1 = P_0 \le P_1 \le \dots \le P_r = G$$ 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.