4. Semidirect products. Let A and B be groups, and let B G A via automorphisms of A. Define A B with respect to this action. (a) Prove that CB(A) = ker, where CB(A) = {b B | ba = ab for all a A} (and ker is the kernel of the action of B on A). (b) Let Z = (*) be the infinite cyclic group. (i) Compute Aut(Z). [Hint. r must map to a generator of Z.] (ii) Classify all actions of Z on itself that correspond to automorphisms (i.e. actions where for each z Z, the map 93: Z - Z defined by a 2.a is an automorphism.) (iii) Classify all semidirect products of Z with itself. [i.e. How many examples are there of ZxZ, which depends intrinsically on the action of Z on itself.] 4. Semidirect products. Let A and B be groups, and let B G A via automorphisms of A. Define A B with respect to this action. (a) Prove that CB(A) = ker, where CB(A) = {b B | ba = ab for all a A} (and ker is the kernel of the action of B on A). (b) Let Z = (*) be the infinite cyclic group. (i) Compute Aut(Z). [Hint. r must map to a generator of Z.] (ii) Classify all actions of Z on itself that correspond to automorphisms (i.e. actions where for each z Z, the map 93: Z - Z defined by a 2.a is an automorphism.) (iii) Classify all semidirect products of Z with itself. [i.e. How many examples are there of ZxZ, which depends intrinsically on the action of Z on itself.]