Coset Spaces

Transversal(G, H): GrpAb, GrpAb -> {\@ GrpAbElt \@}, Map

RightTransversal(G, H): GrpAb, GrpAb -> {\@ GrpAbElt \@}, Map

Given a group \(G\) and a subgroup \(H\) of \(G\), this function returns:

(a)

An indexed set of elements \(T\) of \(G\) forming a right transversal for \(G\) over \(H\); and,

(b)

The corresponding transversal mapping \(\phi: G \rightarrow T\). If \(T = \lbrace t_1, \ldots, t_r\rbrace\) and \(g\) in \(G\), \(\phi\) is defined by \(\phi(g) = t_i\), where \(g\in H t_i\).

Coercions Between Groups and Subgroups

G ! g: GrpAb, GrpAbElt -> GrpAbElt

Given an element \(g\) belonging to the subgroup \(H\) of the group \(G\), rewrite \(g\) as an element of \(G\).

H ! g: GrpAb, GrpAbElt -> GrpAbElt

Given an element \(g\) belonging to the group \(G\), and given a subgroup \(H\) of \(G\) containing \(g\), rewrite \(g\) as an element of \(H\).

K ! g: GrpAb, GrpAbElt -> GrpAbElt

Given an element \(g\) belonging to the group \(H\), and a group \(K\), such that \(H\) and \(K\) are subgroups of \(G\), and both \(H\) and \(K\) contain \(g\), rewrite \(g\) as an element of \(K\).

Morphism(H, G): GrpAb, GrpAb -> ModMatRngElt

The integer matrix defining the inclusion monomorphism from the subgroup \(H\) of \(G\) into \(G\).