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.

Proof:

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 .

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