Soluble Matrix Groups

The functions described in this section apply only to finite groups for which a base and strong generating set may be constructed.

Conversion to a PC-Group

PolycyclicGenerators(G) : GrpMat -> [ GrpPCElt ]

Construct a polycyclic generating sequence for the soluble group G.

PCGroup(G) : GrpMat -> GrpPC, Map

Given a soluble group G, construct a group S in category GrpPC, isomorphic to G. In addition to returning S, the function returns an isomorphism φ: G -> S.

Soluble Group Functions

pCentralSeries(G, p) : GrpMat, RngIntElt -> [ GrpMat ]

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.

p-group Functions

IsSpecial(G) : GrpMat -> BoolElt

Given a p-group G, return true if G is special, false otherwise.

IsExtraSpecial(G) : GrpMat -> BoolElt

Given a p-group G, return true if G is extraspecial, false otherwise.

FrattiniSubgroup(G) : GrpMat -> GrpMat

Given a p-group G, return the Frattini subgroup.

JenningsSeries(G) : GrpMat -> [ GrpMat ]

Given a p-group G, return the Jennings series for G. The series is returned as a sequence of subgroups.

Abelian Group Functions

PrimaryAbelianInvariants(G) : GrpMat -> [ RngIntElt ]

AbelianInvariants(G) : GrpMat -> [ 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.

PrimaryAbelianBasis(G) : GrpMat -> [ GrpMatElt ], [ RngIntElt ]

AbelianBasis(G) : GrpMat -> [ GrpMatElt ], [ RngIntElt ]

Given an abelian group G, return sequences B and I, where I are p-primary invariants for G, and B are generators for G having the orders in I. The non-primary form uses the Smith form invariants, i.e. each element of the sequence divides the next.