By definition, a pcgs determines a pc series of its underlying group. However, in many applications it will be necessary that this pc series refines a normal series with certain properties; for example, a normal series with abelian factors.
There are functions in GAP to compute a pcgs through a normal series with elementary abelian factors, a central series or the lower p-central series. See also Section Special Pcgs for a more explicit possibility.
IsPcgsElementaryAbelianSeries( pcgs ) P
returns true if the pcgs pcgs refines an elementary abelian series.
IndicesNormalSteps gives the indices in the Pcgs, at which normal
subgroups start.
PcgsElementaryAbelianSeries( G ) A
computes a pcgs for G that refines an elementary abelian series.
IndicesNormalSteps gives the indices in the Pcgs, at which normal
subgroups start.
IsPcgsCentralSeries( pcgs ) P
returns true if the pcgs pcgs refines a central series.
IndicesNormalSteps gives the indices in the Pcgs, at which normal
subgroups start.
PcgsCentralSeries( G ) A
computes a pcgs for G that refines a central series.
IndicesNormalSteps gives the indices in the Pcgs, at which normal
subgroups start.
If G is not nilpotent, the result is undefined.
IsPcgsPCentralSeriesPGroup( pcgs ) P
PcgsPCentralSeriesPGroup( G ) A
computes a pcgs for G that refines the lower p-central series.
IndicesNormalSteps gives the indices in the Pcgs, at which normal
subgroups start.
If G is not a p-group, the result is undefined.
PcgsChiefSeries( G ) A
computes a pcgs for G that refines a chief series.
IndicesNormalSteps gives the indices in the Pcgs, at which normal
subgroups start.
Let P = (g1, ¼, gn) be a pcgs such that the pc series determined by P refines a normal series of one of the above types. To obtain the subgroups of this series, the following attributes can be used:
NormalSeriesByPcgs( pcgs ) A
returns the series of all normal subgroups contained in the pc series determined by pcgs.
IndicesNormalSteps( pcgs ) A
Let G be the group defined by pcgs and let pcgs = (g1, ¼, gn). This function returns a sorted list of integers indicating the tails of pcgs which generate a normal subgroup of G. That is, whenever (gi, ¼, gn) is a normal subgroup of G, then i is an element of the list. In particular, the list always starts with 1 and ends with n+1.
gap> g:=Group((1,2,3,4),(1,2));; gap> p:=PcgsElementaryAbelianSeries(g); [ (3,4), (2,4,3), (1,4)(2,3), (1,3)(2,4) ] gap> IndicesNormalSteps(p); [ 1, 2, 3, 5 ] gap> g:=Group((1,2,3,4),(1,5)(2,6)(3,7)(4,8));; gap> p:=PcgsCentralSeries(g); [ (1,5)(2,6)(3,7)(4,8), (5,6,7,8), (5,7)(6,8), (1,4,3,2)(5,6,7,8), (1,3)(2,4)(5,7)(6,8) ] gap> IndicesNormalSteps(p); [ 1, 2, 4, 5, 6 ] gap> q:=PcgsPCentralSeriesPGroup(g); [ (1,5)(2,6)(3,7)(4,8), (5,6,7,8), (5,7)(6,8), (1,4,3,2)(5,6,7,8), (1,3)(2,4)(5,7)(6,8) ] gap> IndicesNormalSteps(q); [ 1, 3, 5, 6 ]
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GAP 4 manual