If G is a group and H a normal subgroup of G, under what conditions is it true that $$H \times (G/H) =G ?$$
To be more specific:
- Characterise groups G which allow proper non-trivial normal subgroups H such that $H \times G/H =G.$
- Given a group G which allow such a decomposition, characterise (normal) subgroups H of G such that $H \times G/H =G.$
Clearly, simple groups allow no such factorisation except the trivial ones.
And the smallest non-Abelian group $S_3$ cannot be the product of its (normal) subgroup $A_3$ and the quotient $\mathbb{Z}_2$.
Also, it follows from the structure theorem for finitely generated Abelian groups that cyclic groups with order a power of a prime do not allow any non-trivial decomposition. It is also not difficult to see that $\mathbb Z$ is not decomposable, i.e., expressible as the product of two proper non-trivial subgroups. These two facts together with the full force of the structure theorem tells us precisely which finitely generated Abelian groups are decomposable, and also enables us to list all the decompositions.
To summarise, in suggestive language, to what extent is the group-quotient operation an inverse for the group-product operation?
