## Two consequences of Frattini’s argument regarding normalizers and centralizers

Let $G$ be a finite group, let $p$ be a prime, let $P \leq G$ be a Sylow $p$-subgroup, and let $N \leq G$ be a normal subgroup such that $p \not| |N|$. Prove the following.

1. $N_{G/N}(PN/N) = N_G(P)N/N$
2. $C_{G/N}(PN/N) = C_G(P)N/N$

Note that $P \cap N = 1$, so that $PN \cong P \times N$. In particular, $P \leq C_G(N)$ and $N \leq C_G(P)$.

Note that $P \leq PN \cong P \times N$ is a Sylow subgroup since $p$ does not divide $|N|$, and that $PN \leq N_G(PN)$ is normal. By Frattini’s Argument, we have $N_G(PN) = PN N_G(P) = N_G(P)N$. Thus $N_G(P)N/N = N_G(PN)/N = N_{G/N}(PN/N)$, as desired.

Now note that $C_G(PN) \leq C_G(P)$, so that $C_{G/N}(PN/N) = C_G(PN)/N \leq C_G(P)N/N$. Moreover, we have $[C_G(P)/N, PN/N] = [C_G(P),PN]/N$ $= [C_G(P),N]/N$ $= 1$. Thus $C_G(P)/N \leq C_{G/N}(PN/N)$.