Algebra Prelim, August 2013

Do all problems

1. Let p be a prime, let H⊲G 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 P∩H 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 (p^m, qⁿ) = R (consider p^m + 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 M≠N) such that M≌N. Prove that there exists a k[x]-module epimorphism M↠C.

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⁺⊗_ℤL^× 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 K∩L is also a finite Galois extension of ℚ.

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