Left transversal of a subgroup

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Definition

Definition with symbols

Let $H$ be a subgroup of a group $G$. Then a subset $S$ of $G$ is termed a left transversal of $H$ in $G$ if $S$ intersects every left coset of $H$ at exactly one element.

$S$ is also termed a system of left coset representatives or set of left coset representatives of $H$ and the elements of $S$ are termed coset representatives of $H$.

Sometimes, the term section is also used for this notion.

Dual notion

The dual notion is that of right transversal of a subgroup.

Algebra loop structure to the left transversal

Consider a subgroup $H$ of a group $G$ and a left transversal $S$ of $H$ in $G$. Then, we can endow $S$ with a binary operation as follows. For $x,y\in S$, we define $x\circ y$ as the left coset representative (with respect to $H$) of $xy$ in $S$. It is easy to see that this gives $S$ the structure of an algebra loop.

When the transversal is a subgroup

If we choose the transversal such that it forms a subgroup, then the algebra loop structure is just the usual group multiplication, so the algebra loop is canonically isomorphic to the subgroup.

When the original subgroup is normal

If the original subgroup is normal, then the algebra loop structure on any left transversal is a group, and this group is isomorphic to the quotient group for that normal subgroup.