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Question 1- Consider the map Φ : (Z, +) →(Zn, + (mod n)) given by Φ(t) ≡ t (mod n).
(a) Prove that Φ is a group homomorphism.
(b) Determine the kernel of Φ.
(c) Determine the image of Φ.
(d) State the first isomorphism theorem and apply it to the map Φ.
Question 2- For any group G, we define the center of G, denoted Z(G), to be Z(G) = {x ∈ G| gx = xg for all g ∈ G}.
(a) Prove that Z(G) is a normal subgroup of G.
(b) Let G be the permutation group S3. Find Z(S3).
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