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 \(\phi: G \ar S\).

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.