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Subsections
Center(G) : GrpPC -> GrpPC
The centre of the group G.
DerivedSubgroup(G) : GrpPC -> GrpPC
DerivedGroup(G) : GrpPC -> GrpPC
The derived subgroup of the group G.
FittingGroup(G) : GrpPC -> GrpPC
The Fitting subgroup of the group G.
The Frattini subgroup of the group G.
Hypercenter(G) : GrpPC -> GrpPC
The hypercentre of the group G, i.e. the stationary term in the
upper central series for G.
pCore(G, S) : GrpPC, RngIntElt -> GrpPC
The maximal normal pi-subgroup of G, O_(pi)(G), where pi is defined by S. The argument
S may be a set of primes, a single prime,
or the negation of a single prime. If S = - p,
then O_(p')(G) is returned.
Given an abelian group G, return sequences B and I such that
( order)(B[i]) = I[i] and < B > = G and the terms
of I give the types of each p-primary component of G.
Invariants(G) : GrpPC -> [RngIntElt]
The abelian invariants of the abelian group G as a sequence of integers.
A chief series for the group G. The series is returned as a sequence
of subgroups of G.
A composition series for the group G. The series is returned as a sequence
of subgroups of G. The i-th term of the composition series has a
presentation given by the generators G.i through G.NPCgens(G) and
relations involving those generators only.
A sequence of integer tuples that describe the composition
factors, ordered according to some composition series
for the group G. Since each factor will be a cyclic
group of prime order, the tuples will each be of
the form <19, 0, q> representing the cyclic group
of order q. The sequence has a custom print routine.
The i + 1-th entry of the composition series for the group G.
Its presentation is given by the generators G.(i + 1) through
G.m, where m is the number of pc-generators of G and
relations involving these generators only.
The derived series of the group G. The series is returned as a
sequence of subgroups.
The derived length of the group G.
An elementary abelian series is a chain of
normal subgroups with the property that the quotient of each pair of
successive terms in the series is elementary abelian.
The elementary abelian series
for the group G is returned as a sequence of subgroups.
The lower central series for the group G. The series is returned
as a sequence of subgroups.
If G is nilpotent, return the nilpotence class of G.
Otherwise, -1 is returned.
The p-central series for G, where p is a prime dividing |G|.
The series is returned as a sequence of subgroups. The p-central series
P_1 triangleright P_2 triangleright ... triangleright P_i
of a soluble group G is defined inductively
as follows:
- P_1 = G,
- P_(i + 1) = (G, P_i)P_i^p, for i > 0.
Given a group G and a subgroup H of G, return a sequence
of subgroups commencing with G and terminating with H, such that
each subgroup is normal in the previous one. If H is not subnormal
in G, the empty sequence is returned.
The upper central series of G. The series is returned as a sequence
of subgroups.
The elementary abelian series of the group D_3 wreath D_5 has terms of
the following orders:
> H := DihedralGroup(GrpPerm, 5);
> G := WreathProduct(DihedralGroup(GrpPC, 3), DihedralGroup(GrpPC, 5),
> [H.2, H.1]);
> EAS := ElementaryAbelianSeries(G);
> for i := 1 to #EAS do
> print FactoredOrder(EAS[i]);
> end for;
[ <2, 6>, <3, 5>, <5, 1> ]
[ <2, 4>, <3, 5>, <5, 1> ]
[ <2, 4>, <3, 5> ]
[ <3, 5> ]
[]
Hence the elementary abelian factors can be seen to have sizes 2^2,
5, 2^4, and 3^5, reading from top to bottom.
The collection of all normal subgroups of G returned as
a sequence.
The lattice of normal subgroups of G.
An elementary abelian minimal normal subgroup of the soluble group G.
Given a non-trivial, normal subgroup N of G, return an elementary
abelian minimal normal subgroup of G contained in N.
Given a normal subgroup N of G, return conjugacy
class representatives of all complements of N in G.
This function implements the first cohomology computation
described in [CNW90].
Given a normal subgroup N of G, return
all normal complements of N in G.
This function implements the first cohomology computation
described in [CNW90].
Given a normal subgroup N of G, and a normal
subgroup H of G containing N,
return all complements of N in H which are
normal in G.
This function implements the first cohomology computation
described in [CNW90].
We define the direct product of an extraspecial group of
order 3^3 and D_3 and let N be the first factor
of this product. Inside the Sylow 3-subgroup, we
see that N has 11 classes of complements, three of
which are normal.
> A := ExtraSpecialGroup(GrpPC,3,1);
> B := DihedralGroup(GrpPC,3);
> G,f,p := DirectProduct(A,B);
> N := f[1](A);
> S3 := Sylow(G,3);
> cS := Complements(S3,N);
> [Index(S3,Normalizer(S3,t)):t in cS];
[ 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3 ]
We can compute only the normal
complements by using NormalComplements.
> ncS := NormalComplements(S3,N);
> #ncS;
3
We can check that precisely one of these three complements
is actually normal in G.
> [IsNormal(G,t):t in ncS];
[ true, false, false ]
Since N has a G-normal complement in S3,
we must have S3 normal in G. We can verify
this. Using the three-parameter version
of NormalComplements we can directly
compute the G-normal complements of N in S3.
> IsNormal(G,S3);
true
> ncG := NormalComplements(G,S3,N);
> #ncG;
1
> #NormalComplements(G,N);
1
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