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

I wonder 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 but I did not take any response.

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

I wonder 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 but I did not take any response.

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 but I did not take any response.

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

I wonder 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 but I did not take any response.