  The notion of normal quantum subgroup introduced in algebraic context by
Parshall and Wang when applied to compact quantum groups is shown to be
equivalent to the notion of normal quantum subgroup introduced by the author.
As applications, a quantum analog of the third fundamental isomorphism theorem
for groups is obtained, which is used along with the equivalence theorem to
obtain results on structure of quantum groups with property F and quantum
groups with property FD. Other results on normal quantum subgroups for tensor
products, free products and crossed products are also proved.
