Different Type of Extension Problem for Groups

[Hello Kitty]

Let H, K be finite groups.

Instead of asking what groups G there are such that K can be embedded as a normal subroup and G/K is isomorphic with H (the usual extension problem for groups), I've been thinking about the following:

Which groups G exist such that H and K can be embedded as (not necessarily normal) subgroups such that G=HK where H and K have trivial intersection?

This neither contains nor is contained in the usual extension problem since there there is no requirement for the normal subgroup to have a complement, and here we don't require any of the factors to be normal.

Clearly for a given G with relevant subgroups identified, there exist cases where we could replace one of the subgroups by a conjugate (e.g. the complement of a semidirect product). Perhaps we could even do the same with an isomorphic non-conjugate. This leads me to think that perhaps some of these 'extensions' should be in some sense 'isomorphic' or 'equivalent'.

I guess my first question before I go any further is is this already an established theory? (It must be.)

 

[fresh_42]

If you start with $H,K$ and build $G=HK$ then $G\cong H \times K$ will do and solve your problem. In all other cases, the real question is how the subgroup operates on the normal subgroup. Any different automorphism results in a different example. The direct product is the trivial automorphism which always works.