I found a proof. Let |G| = sp^e, (p,e) = 1, |P| = p^e. Then p^e | |PN| | sp^e. So

(1) p^e | |PN|

(2) p^(e+1) does not divide |PN|.

From here every thing else follows. For instance |P| = |P int N||PN/N|. Hence |PN| = |P||N:P int N|. If p | |N:P int N|, then p^(e+1) | |PN|, which contradicts (2).