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Image and Preimage of Sylow p-subgroups under Epimorphism
Suppose G and H are p-groups, and $latex \phi:G\to H$ is a surjective homomorphism. Then for any Sylow p-subgroup P of G, $latex \phi(P)$ is a Sylow p-subgroup of H. Conversely, for any Sylow p-sub…