Group having a Sylow tower

Definition
A group having a Sylow tower is a finite group that possesses a defining ingredient::Sylow tower: a defining ingredient::normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, and for each $$p$$ dividing the order of $$G$$, there is a unique quotient that is a $$p$$-subgroup and this group is isomorphic to a $$p$$-Sylow subgroup of $$G$$.

In other words, there exists a normal series:

$$1 = P_0 \le P_1 \le \dots \le P_r = G$$

such that for every $$p$$ dividing the order of $$G$$, there exists a unique $$k$$ such that $$P_k/P_{k-1}$$ is isomorphic to a $$p$$-Sylow subgroup of $$G$$.

Stronger properties

 * Weaker than::Finite nilpotent group

Weaker properties

 * Stronger than::Finite solvable group

Incomparable properties

 * Group having subgroups of all orders dividing the group order: