36.16 Subgroup Series

is a series of normal subgroups of G which cannot be refined further. That is there is no normal subgroup N of G with U_i > N > U_i+1. This attribute returns one chief series (of potentially many possibilities).

gap> g:=Group((1,2,3,4),(1,2));;
gap> ChiefSeries(g);
[ Group( [ (1,2,3,4), (1,2) ] ), 
  Group( [ (2,4,3), (1,4)(2,3), (1,2)(3,4) ] ), 
  Group( [ (1,4)(2,3), (1,2)(3,4) ] ), Group( () ) ]

is a chief series of the group G going through the normal subgroups in the list l. l must be a list of normal subgroups of G contained in each other, sorted by descending size. This attribute returns one chief series (of potentially many possibilities).

returns a series of normal subgroups of G which are invariant under H such that the series cannot be refined any further. G must be a subgroup of H. This attribute returns one such series (of potentially many possibilities).

If U is a subgroup of G this operation returns a subnormal series that descends from G to a subnormal subgroup V ³ U. If U is subnormal, V=U.

gap> s:=SubnormalSeries(g,Group((1,2)(3,4)));
[ Group([(1,2,3,4),(1,2)]),
  Group([(1,2)(3,4),(1,3)(2,4)]),Group([(1,2)(3,4)])]

A composition series is a subnormal series which cannot be refined. This attribute returns one composition series (of potentially many possibilities).

Displays a composition series of G in a nice way, identifying the simple factors.

gap> CompositionSeries(g);
[ Group( [ (3,4), (2,4,3), (1,4)(2,3), (1,2)(3,4) ] ), 
  Group( [ (2,4,3), (1,4)(2,3), (1,2)(3,4) ] ), 
  Group( [ (1,4)(2,3), (1,2)(3,4) ] ), Group( [ (1,2)(3,4) ] ), 
  Group( () ) ]

gap> DisplayCompositionSeries(Group((1,2,3,4,5,6,7),(1,2)));
G (6 gens, size 5040)
 || Z(2)
S (5 gens, size 2520)
 || A(7)
1 (0 gens, size 1)

The derived series of a group is obtained by U_i+1 = U_i¢. It stops if U_i is perfect.

The derived length of a group is the number of steps in the derived series. (As there is always the group, it is the series length minus 1.)

gap> List(DerivedSeriesOfGroup(g),Size);
[ 24, 12, 4, 1 ]
gap> DerivedLength(g);
3

returns a series of normal subgroups of G such that all factors are elementary abelian. If the group is not solvable (and thus no such series exists) it returns fail.

The variant ElementaryAbelianSeriesLargeSteps tries to make the steps in this series large (by eliminating intermediate subgroups if possible) at a small additional cost.

gap> List(ElementaryAbelianSeries(g),Size);
[ 24, 12, 4, 1 ]

For a (solvable) group G and a list of automorphisms morph of G, this command finds a normal series of G with elementary abelian factors such that every group in this series is invariant under every automorphism in morph. If a normal subgroup N of G which is invariant under morph is given, this series is chosen to contain N. No tests are performed to check the validity of the arguments.

gap> g:=Group((1,2,3,4),(1,3));
Group( [ (1,2,3,4), (1,3) ] )
gap> hom:=GroupHomomorphismByImages(g,g,GeneratorsOfGroup(g),
> [(1,4,3,2),(1,4)(2,3)]);
[ (1,2,3,4), (1,3) ] -> [ (1,4,3,2), (1,4)(2,3) ]
gap> InvariantElementaryAbelianSeries(g,[hom]);
[ Group( [ (1,2,3,4), (1,3) ] ), Group( [ (1,3)(2,4) ] ), Group( () ) ]

The lower central series of a group G is defined as U_i+1: = [G,U_i]. It is a central series of normal subgroups. The name derives from the fact that U_i is contained in the i-th step subgroup of any central series.

The upper central series of a group G is defined as an ending series U_i/U_i+1: = Z(G/U_i+1). It is a central series of normal subgroups. The name derives from the fact that U_i contains every i-th step subgroup of a central series.

The p-central series of G is defined by U₁: = G, U_i: = [G,U_i-1]U_i-1^p.

For a p-group G, this function returns its Jennings series. This series is defined by setting G₁ = G and for i ³ 0, G_i+1 = [G_i,G]G_j^p, where j is the smallest integer ³ i/p.

This operation computes the dimensions of the factors of the Loewy series of G. (See Hup82, p. 157 for the slightly complicated definition of the Loewy Series.)

The dimensions are computed via the JenningsSeries without computing the Loewy series itself.

gap> G:= SmallGroup( 3^6, 100 );
<pc group of size 729 with 6 generators>
gap> JenningsSeries( G );
[ <pc group of size 729 with 6 generators>, Group([ f3, f4, f5, f6 ]), 
  Group([ f4, f5, f6 ]), Group([ f5, f6 ]), Group([ f5, f6 ]), 
  Group([ f5, f6 ]), Group([ f6 ]), Group([ f6 ]), Group([ f6 ]), Group([]) ]
gap> DimensionsLoewyFactors(G);
[ 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 27, 
  27, 27, 27, 27, 27, 27, 27, 26, 25, 23, 22, 20, 19, 17, 16, 14, 13, 11, 10,
  8, 7, 5, 4, 2, 1 ]

This function computes an ascending chain of subgroups from U to G. This chain is given as a list whose first entry is U and the last entry is G. The function tries to make the links in this chain small.

This routine tries to find a subgroup E of G, such that G > E > U. If U is maximal, it returns false. This is done by finding minimal blocks for the operation of G on the right cosets of U.

returns a list of all subgroups of G that properly contain U; that is all subgroups between G and U. It returns a record with components subgroups which is a list of these subgroups as well as a component inclusions which lists all maximality inclusions among these subgroups. A maximality inclusion is given as a list [i,j] indicating that subgroup number i is a maximal subgroup of subgroup number j, the numbers 0 and 1+length(subgroups) are used to denote U and G respectively.

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