Invariants of an Abelian Group

ElementaryAbelianQuotient(G, p): GrpAb, RngIntElt -> GrpAb, Map

The maximal \(p\)-elementary abelian quotient of the group \(G\) as GrpAb. The natural epimorphism is returned as second value.

FreeAbelianQuotient(G): GrpAb -> GrpAb, Map

The maximal free abelian quotient of the group \(G\) as GrpAb. The natural epimorphism is returned as second value.

PrimaryAbelianInvariants(A): GrpAb -> [ RngIntElt ]

AbelianInvariants(A): GrpAb -> [ RngIntElt ]

The \(p\)-primary invariants of the abelian group \(A\). Each infinite cyclic factor is represented by zero. The non-primary form gives the Smith form invariants, i.e. each element of the sequence divides the next.

PrimaryAbelianBasis(A): GrpAb -> [ GrpAbElt ], [ RngIntElt ]

AbelianBasis(A): GrpAb -> [ GrpAbElt ], [ RngIntElt ]

Returns sequences \(B\) and \(I\), where \(I\) are the \(p\)-primary invariants of \(A\), and \(B\) are generators for \(A\) with orders as in \(I\). The non-primary form uses the Smith form invariants, i.e. each element of the sequence divides the next.

TorsionFreeRank(A): GrpAb -> RngIntElt

The torsion-free rank of the abelian group \(G\).

TorsionInvariants(A): GrpAb -> [ RngIntElt ]

The torsion invariants of the abelian group \(G\).

pPrimaryInvariants(A, p): GrpAb, RngIntElt -> [ RngIntElt ]

The \(p\)-primary invariants of the abelian group \(G\).