# Left transversal of a subgroup

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## Definition

### Definition with symbols

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

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

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: every subgroup has a left transversal, 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 of a group and a left transversal of in . Then, we can endow with a binary operation as follows. For , we define as the left coset representative (with respect to ) of in . It is easy to see that this gives 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 occurs as a permutable complement to in .

### 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.