Cosets

#matematik-groups

Definition: Coset

Let be a group and be a Subgroup. A left coset of in is a subset of the form where . We define as the set of all left cosets of in .

Right cosets are defined analogously, so and .

The elements of a coset of in are called the coset's representatives.

We quickly see that it doesn't really matter if we are using left or right cosets since they essentially contain the same information. Consider the bijective map . It restricts to a map meaning it defines a bijection .

Theorem

Let be a group and . The following are equivalent:

  1. ,
  2. ,
  3. ,
  4. .

Proof:
(i)(ii): Clear since .
(ii)(iii): Let . Then for some . This gives .
(iii)(iv): We have for some , so .
(iv)(i): For every we have and for every we have .

Corollary

For any group and subgroup , is the disjoint union of all left cosets of in .

Definition: Order, index

Let be a group and be a subgroup. The index of in is and the order of is

Theorem

Let be a group, be a subgroup and . Then there is a bijection; in particular, .

Proof:
Translation by is bijective, hence the result.

Corollary: Theorem of Lagrange

If is a finite group and is a subgroup, then