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.

Existence

The existence of left transversals follows from the axiom of choice, which allows us to pick one representative from each left coset. In fact, the axiom of choice is equivalent to the statement that every subgroup of every group has a left transversal. Further information: existence of left transversals is equivalent to axiom of choice

For a subgroup of finite index, we only need finite choice, which does not require the axiom of choice. For a subgroup of countable index, we only need countable choice, which is a weaker and more tenable assumption than the axiom of choice.

Dual notion

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

A left transversal of a subgroup need not be a right transversal. In fact, a subgroup has the property that every left transversal is a right transversal if and only if it is a normal subgroup.

However, there are many subgroups with the property that there is a left transversal that is also a right transversal. This includes any subgroup of finite group, subgroup of finite index, normal subgroup, retract, permutably complemented subgroup, subset-conjugacy-closed subgroup, and nearly normal 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. If the left transversal is a subgroup, it is also a right transversal, and is a permutable complement to the original subgroup, and the original subgroup is a permutably complemented subgroup.

Note that it is possible for the same subgroup to have multiple non-isomorphic permutable complements. In fact, every group of given order is a permutable complement for symmetric groups. In other words, any group of order $n$ occurs as a permutable complement to $S_{n-1}$ in $S_{n}$.

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.