What's the condition for a group $G$ to be equal to the product of two normal subgroups If $G$ is a group and $N,M$ are two normal subgroups. We know that the product $NM$ is normal subgroup of $G$, but when can I say that $G=NM$. What must be the conditions on $N,M$?
 A: Assuming all groups mentioned in these examples are finite.
An example: if $|G:M|$ and $|G:N|$ are coprime, then $G=NM$. Proof: $|G:NM| \mid |G:M|$ and $|G:NM| \mid |G:N|$. 
Another example: if $|M|$ and $|N|$ are coprime and $|G|=|N| \cdot |M|$, then $G=NM$.
Yet another example: if $M$ is a maximal subgroup and $N \not \subseteq M$, then $G=NM$.
If you are familar with (ordinary) character theory of finite groups: if $\varphi$ is a character of $M$ and the induction-restriction $(\varphi^G)_N$ is irreducible, then $G=NM$.
A: There is a more general question, which has been studied intensively, namely when can we say that $G=AB$ for subgroups $A,B$? Such groups $G$ are called factorizable and there is a large literature about them.
There are some trivial conditions, for example, that $AB$ is a subgroup of $G$ if and only if $AB=BA$, see
Let $A,B$ be subgroups of a group $G$. Prove $AB$ is a subgroup of $G$ if and only if $AB=BA$
References on factorizable groups: for example Arad, and many papers by Amberg,B. Franciosi, S. and Degiovanni and others, also papers by Gorenstein, Herstein.
For more references see also this MO post.
