Question 1: True or false and justify your answer

a. The set Z8\{0} is a group with respect to multiplication.

b. For any binary algebraic structure (S, ) and s, t S, there is a unique solution x S

for the equation x s = t.

c. The group (Z3 Z2, +) is cyclic.

Question 2: Let : G A be an isomorphism between groups (G, ) and (A, ?). If H G, and

(H) = {(h)|h H}, then prove (H) A.

Question 3: Let G be a group, and let w G. Then we define the centralizer of w by Z(w) =

{x G| xw = wx}. Show Z(w) is a subgroup of G.

Please show your work and explanation.