Answer to Question #218521 in Abstract Algebra for nabeel ahmed

Question

If the following are true, give detailed proof. Otherwise, support your answer by a nontrivial example.

(i) S4 is isomorphic to D12.

(ii) H ∼ = K if and only if Aut(H) ∼ = Aut(K).

(iii) Every action of the group G gives the same orbit space.

(iv) The isomorphism class of the multiplicative group of real numbers is non-empty.

(v) The converse of Cauchy’s theorem is true.