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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.
The rank of the Frattini quotient of the p-group G.
Hypercenter(G) : GrpPC -> GrpPC
The hypercentre of the group G, i.e. the stationary term in the
upper central series for G.
A sequence containing all minimal normal subgroups of G.
pCore(G, S) : GrpPC, RngIntElt -> GrpPC
The maximal normal π-subgroup of G, Oπ(G), where π 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 Op'(G) is returned.
The socle of G.
AbelianBasis(G) : GrpPC -> [ GrpPCElt ], [ RngIntElt ]
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.
The non-primary form uses the Smith form invariants, i.e. each element of
the sequence divides the next.
AbelianInvariants(G) : GrpPC -> [RngIntElt]
Given an abelian group G, return a sequence Q containing
the types of each p-primary component of G.
The non-primary form gives the Smith form invariants, i.e. each element of
the sequence divides the next.
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.
Gives a similar result to using ElementaryAbelianSeries, except the
series returned depends only on the isomorphism
type of the group, and consists of characteristic subgroups.
This function may be slower than ElementaryAbelianSeries.
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
P1 triangleright P2 triangleright ... triangleright Pi
of a soluble group G is defined inductively
as follows:
- P1 = G,
- Pi + 1 = (G, Pi)Pip, 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 following functions are only defined for a pc-group which is a
p-group.
Given a p-group G, return the characteristic subgroup of G
generated by the elements xpi, x ∈G, where i is a
positive integer.
Given a p-group G, return the characteristic subgroup of G
generated by the elements of order dividing pi, where i is a
positive integer.
Given a p-group G, return the Jennings series for G. The series is
returned as a sequence of subgroups. The Jennings series
J1 triangleright J2 triangleright ... triangleright Ji ... of a p-group G is defined inductively as follows:
- J1 = G,
- Ji + 1 = <(Ji, G), Jkp>, with k = ⌈(i + 1)/p⌉, i > 0.
The lower exponent-p class of the p-group G.
A sequence whose i-th entry is the number of
pc-generators for the lower exponent-p class i
quotient of the p-group G.
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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