Algebra Prelim, August 2013

Do all problems

1. Let p be a prime, let HG be finite groups, and let P be a subgroup of G. Prove that P is a Sylow p-subgroup of G if and only if PH and PH/H are Sylow p-subgroups of H and G/H respectively.

2. Prove that there is no simple group of order 576 = 9·64.

3. Let R be a UFD with exactly two primes, p and q (i.e. p and q are nonassociate primes, and any prime is an associate of either p or q). Given positive integers m, n, prove that (pm, qn) = R (consider pm + qn). Deduce that R is a PID.

4. Let k be a field, let M be a finitely generated k[x]-module, and let C be a cyclic k[x]-module. Suppose M has a proper submodule N (so MN) such that MN. Prove that there exists a k[x]-module epimorphism MC.

5. Let K and L be finite fields, let K+ indicate the abelian group K under addition, and let L× indicate the abelian group of nonzero elements of L under multiplication. Determine the order of K+L× in terms of | K| and | L|. (You will need to consider two cases, namely whether or not chK divides | L×|.)

6. Let K and L be finite Galois extensions of ℚ. Prove that KL is also a finite Galois extension of ℚ.

7. Let G be a group and let 0→ℤ→PQ→ℤ→ 0 be an exact sequence of G-modules (here ℤ is the trivial G-module), where P and Q are projective G-modules. Prove that H1(G, X)≌H3(G, X) for all G-modules X.

Peter Linnell 2013-08-23