IsOrderedSylowTowerGroup - Maple Help

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GroupTheory

 OrderedSylowTower
 construct a Sylow tower for a finite group
 IsOrderedSylowTowerGroup
 determine if a group is soluble

 Calling Sequence OrderedSylowTower( G, complexion = gamma ) OrderedSylowTower( G ) IsOrderedSylowTowerGroup( G, complexion = gamma ) IsOrderedSylowTowerGroup( G )

Parameters

 G - a permutation group gamma - list of primes including all the prime divisors of the order of G

Description

 • An ordered Sylow tower of complexion $\mathrm{\gamma }$ 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 $G$, and occur in the same order as they do in the list gamma.
 • A finite group may, or may not, have an ordered Sylow tower of a specified complexion. If it has an ordered Sylow tower of one complexion, it may not have an ordered Sylow tower for a different complexion.
 • Every finite nilpotent group has a Sylow tower (of every possible complexion), and a finite group with a Sylow tower (of any complexion) is necessarily soluble.
 • An ordered Sylow tower group is, of course, a Sylow tower group. (See SylowTower.)
 • The OrderedSylowTower( G, 'complexion' = gamma ) command computes an ordered Sylow tower of compexion gamma for the group G if one exists. The returned Sylow tower is an object of type NormalSeries.
 • 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 IsOrderedSylowTowerGroup( G, 'complexion' = gamma ) command returns true if G has a Sylow tower of complexion gamma, and returns false if not.
 • Both OrderedSylowTower and IsOrderedSylowTowerGroup can be called without the complex = gamma option, in which case the default complexion used is the list of all the prime divisors of the order of the group G in descending order. (An ordered Sylow tower of this complexion is sometimes called an ordered Sylow tower of supersoluble type.)

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{seq}\left(H,H=\mathrm{OrderedSylowTower}\left(G\right)\right)$
 > $T≔\mathrm{OrderedSylowTower}\left(G,'\mathrm{complexion}'=\left[2,3\right]\right)$
 ${T}{≔}⟨⟩{▹}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{▹}⟨\left({1}{,}{4}{,}{3}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩$ (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)
 > $\mathrm{IsOrderedSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsOrderedSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right),'\mathrm{complexion}'=\left[5,2\right]\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{IsOrderedSylowTowerGroup}\left(\mathrm{DihedralGroup}\left(5\right),'\mathrm{complexion}'=\left[2,5\right]\right)$
 ${\mathrm{false}}$ (9)

Compatibility

 • The GroupTheory[OrderedSylowTower] and GroupTheory[IsOrderedSylowTowerGroup] commands were introduced in Maple 2019.