Suppose that $G=MN$ and $G=MP$ are two exact factorization of a finite group $G$. What is the relation between $M$ and $P$? Clearly if $G=MN$ then $G=M(mNm^{-1})$ is another factorization of $G$. Is this the only possibility to change $N$ into $P$?



Exact factorization of finite groups
The answer of the problem is the following:


A group G has two exact factorizations $G = M N = M P$ if and only if there exists a **unitary bijective map** (just a map, not a morphism of groups) $v : N \to P$ such that $n v(n)^{-1} \in M$, for all $n \in N$ and **$v$ satisfy four *natural* compatibility conditions** (not very transparent to write down here -- they are similar, mutatis mutandis, to the one given in Propositions 2.1 in <https://arxiv.org/abs/0903.5060> -- for example one of them is that v is a morphism of $M$-sets).


