Algebra Prelim Fall 1994   Answer All Problems

1.
Let p be a prime, let G be a finite p-group, let Z be the center of G, and let 1H G.

(i) Let x HZ, and let (x) denote the conjugacy class containing x. Prove that (x) H and p divides |(x)|.

(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.)

2.
Let G be a simple group of order 180.

(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.

3.
Let R be a PID (principal ideal domain), and let M be a cyclic left R-module. Suppose M = A B where A and B are nonzero left R-modules. Prove that there exists r R 0 such that rM = 0. Prove further that for such an r, there exist distinct primes p, q R such that pq divides r.

4.
Let R be the ring [[X]]/(X - 2), where [[X]] denotes the power series ring in X over , the localization of at the prime 2.

(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).

5.
Let p be a prime, let K L be fields of characteristic p, let , L, and let d be a positive integer. Suppose [K() : K] = d, [K() : K] = p, is separable over K, and is not separable over K.

(i) Prove that K() = K() and K.

(ii) Prove that K() K( + ).

(iii) Prove that [K( + ) : K] = pd.

6.
Let p be a prime, and let K = (t), the quotient field (field of fractions) of the polynomial ring [t].

(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.

Peter Linnell
1998-06-07