Let be a finite group, let be a prime, let be a Sylow -subgroup, and let be a normal subgroup such that . Prove the following.
Note that , so that . In particular, and .
Note that is a Sylow subgroup since does not divide , and that is normal. By Frattini’s Argument, we have . Thus , as desired.
Now note that , so that . Moreover, we have . Thus .