A Sylow tower for a finite group $G$ is a normal series

such that, for each $i$, the quotient group $\frac{G_i}{G_{i+1}}$ is isomorphic to a Sylow $p_i$-subgroup of $G$, for some prime $p_i$, and such that $p_1$, $p_2$, ..., $p_r$  are all the distinct prime divisors of the order of $G$.

A finite group may, or may not, have a Sylow tower. If such a Sylow tower exists, then the ordered sequence $p_1,p_2,..,p_r$ of primes (or any ordered sequence of prime numbers containing it in the same order) is called the complexion of the Sylow tower.

Every finite nilpotent group has a Sylow tower (of arbitrary complexion), and a finite group with a Sylow tower is necessarily soluble.

A group with a Sylow tower need not have an ordered Sylow tower. (See OrderedSylowTower.)

The SylowTower( G ) command computes a Sylow tower for the group G if one exists. The returned Sylow tower is an object of type NormalSeries. Since the terms of a Sylow tower are not computed until they are needed (such as for printing or accessing them to compute the complexion), if a group has not Sylow tower then an exception is not raised until the terms are accessed.

In addition to the methods available for any Series object, a Sylow tower T also supports the Complexion( T ) method, which returns the complexion of the computed tower, as a list of primes.

The IsSylowTowerGroup( G ) command returns true if G has a Sylow tower (of some complexion), and returns false if no Sylow tower of any complexion exists.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$T≔\mathrm{SylowTower}\left(G\right)$

$T≔⟨⟩◃⟨\left(1\,2\right)\left(3\,4\right)\,\left(1\,3\right)\left(2\,4\right)⟩◃{\mathbf{A}}_4$

$\mathrm{type}\left(T\,'\mathrm{NormalSeries}'\right)$

$\mathrm{true}$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(H\right)\,H=T\right)$

$1,4,12$

$\mathrm{seq}\left(\mathrm{Index}\left(T\left[i-1\right]\,T\left[i\right]\right)\,i=2..:-\mathrm{numelems}\left(T\right)\right)$

$4,3$

$\mathrm{Complexion}\left(T\right)$

$\left[2\,3\right]$

$T≔\mathrm{SylowTower}\left(\mathrm{SigmaL}\left(3\,3\right)\right)\:$

$\mathrm{Complexion}\left(T\right)$

Error, group has no Sylow tower

$\mathrm{IsSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$

$\mathrm{true}$

$\mathrm{IsSylowTowerGroup}\left(\mathrm{AGL}\left(4\,4\right)\right)$

$\mathrm{false}$

The GroupTheory[SylowTower] and GroupTheory[IsSylowTowerGroup] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.