Let G=HK, where H, K are characteristic subgroups of G with trivial intersection, i.e. . Then, .
Now suppose , where and are characteristic subgroups of with . Define by
is a homomorphism, and bijective since . Thus and similarly, so that is well-defined.
Note that so is a homomorphism.
Suppose . Then . Then for , so that . Thus is injective.
For any , define . Then
So is a homomorphism.
If , then , so that . Then since , so , so that . Thus and is injective.
Any can be written as since is bijective. Similarly, any can be written as . Then so is surjective.
Thus . Note that since . Similarly, . So and is surjective.
Hence is an isomorphism.