What is the Significance of the Orbit of P in Sylow's Theorems?

[mathsss2]

Let p be a prime, G a finite group, and P a p-Sylow subgroup of G. Let M be any subgroup of G which contains N_G(P). Prove that [G:M]\equiv 1 (mod p). (Hint: look carefully at Sylow's Theorems.)

 

[mathwonk]

since the word N = normalizer occurs, one is led to look at the action on G on p by conjugation. G permutes subgroups of G and we look at the orbit of P. this orbit contains kp+1 subgroups, so the theorem holds if N = M. then what?