Direct product and group generated by subgroups

Question

Lets have two subgroups $A$ and $B$ of given group $G$. We have their product defined the following way:
$$AB=\{ab\,|\, a\in A,\, b\in B \}$$
But, unfortunately, this product is not always a group.
Suppose we have another set $C$ that is defined the following way:
$$C=\{ab\,|\, (a\in A \lor a\in B) \land (b\in A \lor b\in B) \}$$
Obviously, $C$ is a group because $C \cong A \times B$ and it is a subgroup of $G$.
The question is: whether there exists a math term for this subgroup product so I could find more information about in order to figure out the proof for Cauchy theorem by myself.

Answer 1

The "math term" for $G=AB$ is factorizable groups. There is a large literature on such groups. One open question is, whether or not any dinilpotent group $G=AB$, which is the product of two nilpotent subgroups $A$ and $B$, is solvable - see this question.

By the way, $AB$ is a group if and only if $AB=BA$:

Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$