Interaction of Sylow subgroups with quotients

Let p be a prime and let G be a finite group of order p^am, where p does not divide m. Let P \leq G be a subgroup of order p^a and N \leq G a normal subgroup of order p^bn where p does not divide n. Prove that |P \cap N| = p^b and that |PN/N| = p^{a-b}.


By the Second Isomorphism Theorem, we have PN \leq G, N \leq PN normal, P \cap N \leq P normal, and P/(P \cap N) \cong PN/N. Now |PN| divides |G| by Lagrange, so that |PN| = p^k\ell for some k where p does not divide \ell; then \ell|m. Because P \leq PN, we have k = a, and because N \leq PN, n|\ell. Thus |PN/N| = p^{a-b}q, where p does not divide q. Note that |P/(P \cap N)| = p^k for some k, and we have p^k = p^{a-b}q. Thus q = 1 and we have |PN/N| = p^{a-b}. Finally, we have |P|/|P \cap N| = p^a / p^b, so that |P \cap N| = p^b.

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