## Quotient Groups

Ifis a normal subgroup ofthen the coset spacewith the binary operation

for allbelonging tois a group. The identity element of this group is the trivial coset

Let G be a group and let H be a subgroup of G. The following statements are equivalent:

is a normal subgroup of

For all(The left and right cosets are identical)

Coset multiplication is well-defined that is ifandthenOnce this is established the rule for multiplication of cosets follows.

*Proof*

Ifis normal inand andthenso

Supposefor allSupposeThen

Suppose coset multiplication is well defined. Letsoand sois normal in

Now the main result: Ifis a normal subgroup ofthe set of cosetsbecomes a group under coset multiplication.

*Proof*

For associativity, note that

and

Hence,is the identity for coset multiplication.

Finallysoforand every coset (element of the sethas an inverse henceis a group, called the quotient group ofby