Group having a Sylow tower

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
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Definition

A group having a Sylow tower is a finite group that possesses a Sylow tower: a 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:

$1=P_{0}\leq P_{1}\leq \dots \leq P_{r}=G$

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 $G$.

Relation with other properties

Stronger properties

• Finite nilpotent group

Weaker properties

• Finite solvable group

Incomparable properties

• Group having subgroups of all orders dividing the group order: For full proof, refer: Sylow tower not implies subgroups of all orders dividing the group order, Subgroups of all orders dividing the group order not implies Sylow tower

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 258, Chapter 7 (Fusion, transfer and p-factor groups), Section 7.6 (Elementary applications), ^{More info}