IsSylowTowerGroup - Maple Help

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GroupTheory

 SylowTower
 construct a Sylow tower for a finite group
 IsSylowTowerGroup
 determine if a group has a Sylow tower

 Calling Sequence SylowTower( G ) IsSylowTowerGroup( G )

Parameters

 G - a permutation group

Description

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

$1={G}_{0}◃{G}_{1}◃\dots ◃{G}_{r}=G$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $T≔\mathrm{SylowTower}\left(G\right)$
 ${T}{≔}⟨⟩{◃}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{◃}{{\mathbf{A}}}_{{4}}$ (2)
 > $\mathrm{type}\left(T,'\mathrm{NormalSeries}'\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(H\right),H=T\right)$
 ${1}{,}{4}{,}{12}$ (4)
 > $\mathrm{seq}\left(\mathrm{Index}\left(T\left[i-1\right],T\left[i\right]\right),i=2..:-\mathrm{numelems}\left(T\right)\right)$
 ${4}{,}{3}$ (5)
 > $\mathrm{Complexion}\left(T\right)$
 $\left[{2}{,}{3}\right]$ (6)

Observe that no exception is raised here, since the Sylow tower object is not printed and so the terms are not yet computed.

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

Attempting to access the terms of the Sylow tower object T causes an exception to be raised.

 > $\mathrm{Complexion}\left(T\right)$
 > $\mathrm{IsSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsSylowTowerGroup}\left(\mathrm{AGL}\left(4,4\right)\right)$
 ${\mathrm{false}}$ (8)

Compatibility

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