Some functions described in this section may not exist or may have restrictions for some categories of groups. Details can be found in the chapters on the individual categories.
Construct the centre of the group G.
Construct the hypercentre of the group G (the stationary term of the upper central series).
The derived length of G. If G is non-soluble, the function returns the number of terms in the series terminating with the soluble residual.
The derived series of the group G. The series is returned as a sequence of subgroups.
The derived subgroup of the group G.
The Fitting subgroup of the group G.
Given a group G that is a p-group, return the Frattini subgroup.
Given a p-group G, return the Jennings series for G. The series is returned as a sequence of subgroups.
The lower central series of G. The series is returned as a sequence of subgroups.
The nilpotency class of the group G. If the group is not nilpotent, the value -1 is returned.
The normal closure of the subgroup H of group G.
The normal subgroups of G arranged as a lattice.
The normal subgroups of G.
Given a soluble group G, and a prime p dividing |G|, return the lower p-central series for G. The series is returned as a sequence of subgroups.
The maximal normal solvable subgroup of the group G.
The solvable residual of the group G.
Given a group G and a subnormal 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 commencing with the trivial subgroup. Since the algorithm used requires the conjugacy classes of G, this function is much more restricted in its range of application than DerivedSeries and LowerCentralSeries.
Given a finite group G in the category GrpPerm, GrpMat,
GrpPC of GrpAb, return a sequence S of tuples that
represent the composition factors of G, ordered according to some
composition series of G. Each tuple is a triple
of integers f, d, q that defines the isomorphism type of
the corresponding composition factor. A triple < f, d, q >
describes a simple group as follows. The integer f defines
the family to which the group belongs, and d and q are the
parameters of the family. The length of the sequence S is the
number of composition factors of G.
The families are:
f family name
-------------------------
1 A(d, q)
2 B(d, q)
3 C(d, q)
4 D(d, q)
5 G(2, q)
6 F(4, q)
7 E(6, q)
8 E(7, q)
9 E(8, q)
10 2A(d, q)
11 2B(2, q)
12 2D(d, q)
13 3D(4, q)
14 2G(2, q)
15 2F(4, q)
16 2E(6, q)
17 Alternating(d)
18 Sporadic group --- see next list
19 Cyclic(q)
For f=18, the sporadic groups are:
d group name
-------------------------
1 M_11
2 M_12
3 M_22
4 M_23
5 M_24
6 J_1
7 HS
8 J_2
9 McL
10 Suz
11 J_3
12 Co_1
13 Co_2
14 Co_3
15 He
16 Fi_22 = M(22)
17 Fi_23 = M(23)
18 Fi_24 = M(24)
19 Ly
20 Ru
21 ON
22 Th
23 HN
24 BM
25 M
26 J_4
Given an abelian group G in the category GrpPerm, GrpMat, GrpPC or GrpAb, 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..
Given an abelian group G in the category GrpPerm, GrpPC of GrpAb, return sequences B and I where I contains the types of each p-primary component of G and B contains corresponding elements of G which have the order given and generate G. The non-primary form uses the Smith form invariants, i.e. each element of the sequence divides the next.