construct a composition series of a finite group
compute the composition length of a group
CompositionSeries( G )
CompositionLength( G )
a permutation group
A composition series of a group G is a subnormal series
of G, for which each term is a maximal normal subgroup in the preceding term, so that the successive quotients GkGk+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 GkGk+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.
G ≔ Alt⁡4
cs ≔ CompositionSeries⁡DihedralGroup⁡8
G ≔ Group⁡Perm⁡1,2,3,Perm⁡1,2,Perm⁡4,5,6,Perm⁡4,5,Perm⁡7,8,9,Perm⁡1,4,7,2,5,8,3,6,9,Perm⁡1,4,2,5,3,6
G≔ < a permutation group on 9 letters with 7 generators >
cs ≔ CompositionSeries⁡G
cs≔ < a permutation group on 9 letters with 7 generators > ▹ < a permutation group on 9 letters with 7 generators > ▹…▹1,2,37,8,9▹
The GroupTheory[CompositionSeries] and GroupTheory[CompositionLength] commands were introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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