5. G = {a + b2: a eZ and be Z) is a group under addition in the reals. Define o: G G for all a+ b./2 eG, as (a+ b /2) = a- b./2. Prove that o is an automorphism.

Show that the mapping Φ(a + bi) = a - bi is an automorphism of the group of complex numbers under addition. Show that Φ preserves complex multiplication as well—that is, Φ(xy) = Φ(x)Φ(y)

Let (IR,+) be a group of real numbers under addition and (R+,-) be the group of positive real numbers under multiplication. Prove f: R→ R+ by f (x)= ex for all x ER is homomorphism and isomorphism.

6. Additional Classifications: Classify the factor group according to the fundamental theorem of finitely generated abelian groups: a. (Z₁ × Z₁2)/((3,6)) b. (Z x Z)/((3,6))

I need help with #4.20

If ring (R1, +, *) is isomorphic to ring (R2, +, *) with f being the isomorphism,then prove the following statements:a. f(e1) = e2, where e1 is the identity element for the group (R1, +) and e2 isthe identity element of the group (R2, +).b. f(-a) = -f(a), for each element ‘a’ in R1, where –a represent the additiveinverse of element ‘a’ in R1 and –f(a) represents the additive inverse of theelement f(a) in R2.c. If (R1, +, *) is a ring with (multiplicative) identity I1 and (R2, +, *) is a ringwith (multiplicative) identity I2, then f(I1) = I2.