Consider the map phi z rarrzn mod n given by phit equiv

Question

Question 1- Consider the map Φ : (Z, +) →(Z_n, + (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).