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).

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