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I am having trouble with this question so it would be really nice if anyone could provide some help.

Let $$\phi: G \to G'$$ be a group homomorphism, and let $$H' \le G'$$ be a subgroup of G'.

a) Show that $$H=\phi^{-1}(H')$$ is a subgroup of G.

b) Now suppose H’ is a normal subgroup of G’. Does it follow that $$H=\phi^{-1}(H’)$$ is a normal subgroup of G?

c) Now suppose phi is surjective (but not that H’ is normal). Show that there is a bijective correspondence between $$K’ \le G’$$ containing H' and $$K \le G$$ containing H.

Thanks