**Theorem**:Let $P\in Syl_p(G)$ for a finite group $G$. Then $G$ has a normal $p-$ complement if and only if $P$ conrols its own fusion.

I wonder that if similar argument is true for Hall subgroups (in general or in solvable groups), if yes, Is there any reference ?  

I had asked it [there][1] but I did not take any response.


  [1]: http://math.stackexchange.com/questions/1974798/a-hall-subgroup-controls-its-own-fusion