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

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.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$\mathrm{CompositionSeries}\left(G\right)$

${\mathbf{A}}_4▹\left[{\mathbf{A}}_4\,{\mathbf{A}}_4\right]▹⟨\left(1\,3\right)\left(2\,4\right)⟩▹⟨⟩$

$\mathrm{CompositionLength}\left(G\right)$

Warning, over-writing property `["DerivedSeries"]' with a different value

$3$

$\mathrm{IsSimple}\left(\mathrm{PSL}\left(3\,3\right)\right)$

$\mathrm{true}$

$\mathrm{CompositionSeries}\left(\mathrm{PSL}\left(3\,3\right)\right)$

$\mathbf{PSL}\left(3\,3\right)▹⟨⟩$

$\mathrm{CompositionLength}\left(\mathrm{PSL}\left(3\,3\right)\right)$

$1$

$\mathrm{cs}≔\mathrm{CompositionSeries}\left(\mathrm{DihedralGroup}\left(8\right)\right)$

$\mathrm{cs}≔{\mathbf{D}}_8▹⟨\left(1\,7\,5\,3\right)\left(2\,8\,6\,4\right)\,\left(1\,2\,3\,4\,5\,6\,7\,8\right)⟩▹\dots ▹⟨\left(1\,5\right)\left(2\,6\right)\left(3\,7\right)\left(4\,8\right)⟩▹⟨⟩$

$\mathrm{type}\left(\mathrm{cs}\,'\mathrm{SubnormalSeries}'\right)$

$\mathrm{true}$

$\mathrm{type}\left(\mathrm{cs}\,'\mathrm{NormalSeries}'\right)$

$\mathrm{false}$

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

$16,8,4,2,1$

$G≔\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[4\,5\,6\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[4\,5\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[7\,8\,9\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,4\,7\right]\,\left[2\,5\,8\right]\,\left[3\,6\,9\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,4\right]\,\left[2\,5\right]\,\left[3\,6\right]\right]\right)\right]\right)$

$G≔⟨\left(1\,2\,3\right)\,\left(1\,2\right)\,\left(4\,5\,6\right)\,\left(4\,5\right)\,\left(7\,8\,9\right)\,\left(1\,4\,7\right)\left(2\,5\,8\right)\left(3\,6\,9\right)\,\left(1\,4\right)\left(2\,5\right)\left(3\,6\right)⟩$

$\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$

$\mathrm{cs}≔⟨\left(1\,2\,3\right)\,\left(1\,2\right)\,\left(4\,5\,6\right)\,\left(4\,5\right)\,\left(7\,8\,9\right)\,\left(1\,4\,7\right)\left(2\,5\,8\right)\left(3\,6\,9\right)\,\left(1\,4\right)\left(2\,5\right)\left(3\,6\right)⟩▹⟨\left(4\,5\right)\left(7\,8\right)\,\left(5\,6\right)\left(7\,9\right)\,\left(1\,3\,2\right)\,\left(2\,3\right)\left(7\,8\right)\,\left(1\,4\,7\right)\left(2\,5\,8\right)\left(3\,6\,9\right)\,\left(8\,9\right)⟩▹\dots ▹⟨\left(1\,2\,3\right)\left(4\,5\,6\right)⟩▹⟨⟩$

$\mathrm{CompositionLength}\left(G\right)$

Warning, over-writing property `["DerivedSeries"]' with a different value

$8$

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

$1296,648,324,108,54,27,9,3,1$

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

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