Subsections

• Characteristic Subgroups and Subgroup Series
• The Abstract Structure of a Group

Center(G) : GrpFin -> GrpFin

Construct the centre of the group G.

Hypercenter(G) : GrpFin -> GrpFin

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.

DerivedGroup(G) : GrpFin -> GrpFin

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.

NormalClosure(G, H) : GrpFin, GrpFin -> GrpFin

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.

SolvableResidual(G) : GrpFin -> GrpFin

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.

Invariants(G) : GrpFin -> [ RngIntElt ]

Given an abelian group G, return a sequence Q containing the types of each p-primary component of G.

Given an abelian group G, 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.

Given a group G, 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      M(22)
   17      M(23)
   18      M(24)
   19      LY
   20      RU
   21      ON
   22      TH
   23      HA
   24      BM
   25      M
   26      J_4