Use proposition 13: |PN|= |P||N| / |P intersect N| isomorphism theorem

7. Let p be a prime and let G be a group of order pom, where p does not divide m. Let P be a subgroup of G of order pa and let N be a normal subgroup of G of order pon, where p does not divide n. Prove that |Pn N| = p and | PN/N| = pa-b. Note: The subgroup P of G is called a Sylow p-subgroup of G. This exercise shows that the intersection of a Sylow p-subgroup of G with any normal subgroup N of G is a Sylow p-subgroup of N