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.
Ifis normal inand andthenso
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.
For associativity, note that
Hence,is the identity for coset multiplication.
Finallysoforand every coset (element of the sethas an inverse henceis a group, called the quotient group ofby