Given a group G
- How can one tell if it could have been formed as the direct product of sub groups.
- If so what are the groups.
Given a group G
Here is a `simple' procedure.
Enumerate all non-trivial normal subgroups, $N$, of $G$.
Check if the extension $N \stackrel{f}{\to} G \stackrel{h}{\to} G/N$ is split; ie there exists some homomorphism $s : G/N \to G$ such that $s h = id_{G/N}$.
Finally, show that the image of $s$ is a normal subgroup of $G$.
The hard part of this whole process is step 2, which basically requires you to figure out some way of picking a representative element out of each of the cosets of $G/N$. If the extension is indeed split, there is really only one way to do this (ie map each element $x \in G/N$ to the `pair' $(x, e)$ where $e$ is the identity of $N$); and if it is not split then it is impossible.