Abstract Algebra Answers

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

Let G be a nonabelian group of order 10 having golden ratio 1+√5 2 as the neutral element. Then:

(a) Verify class equation for G.

(b) Find all elements of Inn(G).

(c) Verify that G/Z(G) ∼ = Inn(G).

(d) For some elements x,y,z ∈ G: verify the commutator identity: [xy,z] = [x,z][x,z,y][y,z]