CompositionLength - Maple Help

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GroupTheory

 CompositionSeries
 construct a composition series of a finite group
 CompositionLength
 compute the composition length of a group

 Calling Sequence CompositionSeries( G ) CompositionLength( G )

Parameters

 G - a permutation group

Description

 • A composition series of a group $G$ is a subnormal series

$G={G}_{0}▹{G}_{1}▹\dots ▹{G}_{r}=1$

of $G$, for which each term is a maximal normal subgroup in the preceding term, so that the successive quotients $\frac{{G}_{k}}{{G}_{k+1}}$ are simple groups.

 • Every finite group has a composition series, and any two composition series for a finite group have the same number of terms, and the multi-set of isomorphism types of the quotients $\frac{{G}_{k}}{{G}_{k+1}}$ is unique (apart from order).  The number $r$ of terms in a composition series is therefore independent of the chosen series, and so the composition length, $r-1$ of the group $G$ is well-defined.
 • The CompositionSeries( G ) command constructs a composition series of a finite group G. The group G must be an instance of a permutation group. The returned composition series of G is represented by a series data structure which admits certain operations common to all series.  See GroupTheory[Series].
 • The CompositionLength( G ) command returns the composition length of G; that is, the length of a composition series of G. This is the number of subgroup inclusions - so it is one less than the number of groups in the derived series.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $\mathrm{CompositionSeries}\left(G\right)$
 ${{\mathbf{A}}}_{{4}}{▹}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{▹}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right)⟩{▹}⟨⟩$ (2)
 > $\mathrm{CompositionLength}\left(G\right)$
 ${3}$ (3)
 > $\mathrm{IsSimple}\left(\mathrm{PSL}\left(3,3\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{CompositionSeries}\left(\mathrm{PSL}\left(3,3\right)\right)$
 ${\mathbf{PSL}}\left({3}{,}{3}\right){▹}⟨⟩$ (5)
 > $\mathrm{CompositionLength}\left(\mathrm{PSL}\left(3,3\right)\right)$
 ${1}$ (6)
 > $\mathrm{cs}≔\mathrm{CompositionSeries}\left(\mathrm{DihedralGroup}\left(8\right)\right)$
 ${\mathrm{cs}}{≔}{{\mathbf{D}}}_{{8}}{▹}⟨\left({1}{,}{7}{,}{5}{,}{3}\right)\left({2}{,}{8}{,}{6}{,}{4}\right){,}\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right)⟩{▹}{\dots }{▹}⟨\left({1}{,}{5}\right)\left({2}{,}{6}\right)\left({3}{,}{7}\right)\left({4}{,}{8}\right)⟩{▹}⟨⟩$ (7)
 > $\mathrm{type}\left(\mathrm{cs},'\mathrm{SubnormalSeries}'\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{type}\left(\mathrm{cs},'\mathrm{NormalSeries}'\right)$
 ${\mathrm{false}}$ (9)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(H\right),H=\mathrm{cs}\right)$
 ${16}{,}{8}{,}{4}{,}{2}{,}{1}$ (10)
 > $G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[4,5,6\right]\right]\right),\mathrm{Perm}\left(\left[\left[4,5\right]\right]\right),\mathrm{Perm}\left(\left[\left[7,8,9\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,4,7\right],\left[2,5,8\right],\left[3,6,9\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,4\right],\left[2,5\right],\left[3,6\right]\right]\right)\right]\right)$
 ${G}{≔}{\mathrm{< a permutation group on 9 letters with 7 generators >}}$ (11)
 > $\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$
 ${\mathrm{cs}}{≔}{\mathrm{< a permutation group on 9 letters with 7 generators >}}{▹}{\mathrm{< a permutation group on 9 letters with 7 generators >}}{▹}{\dots }{▹}⟨\left({1}{,}{2}{,}{3}\right)⟩{▹}⟨⟩$ (12)
 > $\mathrm{CompositionLength}\left(G\right)$
 ${8}$ (13)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(H\right),H=\mathrm{cs}\right)$
 ${1296}{,}{648}{,}{324}{,}{108}{,}{54}{,}{27}{,}{9}{,}{3}{,}{1}$ (14)

Compatibility

 • The GroupTheory[CompositionSeries] and GroupTheory[CompositionLength] commands were introduced in Maple 2019.