Suppose G and H are p-groups, and is a surjective homomorphism.
Then for any Sylow p-subgroup P of G, is a Sylow p-subgroup of H.
Conversely, for any Sylow p-subgroup Q of H, for some Sylow p-subgroup P of G.
By the First Isomorphism Theorem, . Write . Then .
Since is a Sylow -subgroup of , is relatively prime to . Thus, is also relatively prime to .
Then is also relatively prime to . Since , is a -group, so is a Sylow -subgroup of .
Part 2: Let be a Sylow -subgroup of . Then by Correspondence Theorem, for some subgroup with .
Then, is relatively prime to , so contains a Sylow -subgroup .
Consider . By previous part, is a Sylow -subgroup of , so .