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Let G and H be groups and f:G--->H be a surjective homomorphism. Let J be a...

Let G and H be groups and f:G--->H be a surjective homomorphism. Let J be a subgroup of H and define f^-1(J) ={x is an element of G| f(x) is an element of J}

a. Show ker(f)⊂f^-1(J) and ker(f) is a normal subgroup of f^-1(J)

b. Let p: f^-1(J) --> J be defined by p(x) = f(x). Show p is a surjective homomorphism

c. Show the set kef(f) and ker(p) are equal

d. Show J is isomorphic to f^-1(J)/ker(f)

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