# 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 $G$.