Refer to Exercises 29 and 30 for the following questions. 

a. How many elements are there in Z₂^Z2 ? in Z₃^Z3 

b. Classify (Z₂^z2 , +) and (Z₃^z3 , +) by Theorem 11.12, the Fundamental Theorem of finitely generated abelian groups. 

c. Show that if F is a finite field, then F^F= P_F.

Data from Exercise 29

Let R be a ring, and let R^R be the set of all functions mapping R into R. For∅, ψ ∈ R^R, define the sum∅ + ψ by (∅ + ψ)(r) = ∅(r) + ψ(r) and the product ∅ . ψ by (∅ • ψ)(r) = ∅(r)ψ(r)  for r ∈ R. Note that · is not function composition. Show that (R^R,+,•) is a ring.