- a) Consider f: Z × Z → Z defined by ƒ ((a, b)) = a − b. Show that f is a surjective homomorphism. [Recall: Z is an additive group. So is ZxZ] b) What is the identity element of Z? c) Find the kernel of f. d) Is the kernel off cyclic? If so, find the element (a, b) = Z× Z so that < (a, b) > is equal to the kernel of f. e) Can you apply the First Isomorphism Theorem to ƒ? Why? f) If possible, apply the First Isomorphism Theorem to f.

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