(ii) Prove that Z H1.
(iii) Let A be a maximal normal abelian subgroup of G. Prove that A is also a maximal abelian subgroup of G. (Apply (ii) with G = G/A and H the centralizer of A in G.)
(i) Prove that the number of 5-Sylow subgroups of G is 36.
(ii) Prove that the normalizer of a 3-Sylow subgroup of G has order 18.
(iii) Prove that the 3-Sylow subgroup of a group of order 18 is normal in that group.
(iv) If A and B are distinct 3-Sylow subgroups of G, prove that A B = 1 (consider the centralizer in G of A B).
(v) Prove that there is no simple group of order 180.
(i) If q is an odd integer, prove that q is invertible in [[X]]/(X - 2).
(ii) Define : [[X]][[X]] by aiXi = aiXi, and let : [[X]]R be the natural epimorphism. Prove that is surjective and deduce that R [[X]]/(X - 2).
(iii) Prove that R [X]/(X - 2).
(i) Prove that K() = K() and K.
(ii) Prove that K() K( + ).
(iii) Prove that [K( + ) : K] = pd.
(i) Prove that Xp - t is irreducible in K[X].
(ii) Let L be the splitting field of Xp - t over K. Determine the Galois group of L over K.