Algebra Prelim, August 2012

Do seven problems

1. Let G be a simple group of order 4312 = 23·72·11.
(a)
Prove that G has a subgroup of order 77 (consider the number of Sylow 11-subgroups).

(b)
Prove that G has a subgroup of order 7 whose normalizer contains a group of order 49 and a group of order 77.

(c)
Prove that no such G exists.

2. Let R be a UFD and let I1I2⊆...⊆R be an ascending chain of principal ideals of R. Prove there exists a positive integer N such that In = IN for all nN.

3. Let q∈ℤ be a prime and let P be a nonzero projective ℤ-module (not necessarily finitely generated).
(a)
Prove that PPq.

(b)
Prove there exists a ℤ-module epimorphism P↠ℤ/qℤ (you may assume that every subspace of a vector space has a direct complement).

(c)
Prove that PP≠ 0.

4. Let M and N be finitely generated ℤ-modules. Suppose M is isomorphic to a submodule of N and N is isomorphic to a submodule of M. Prove that MN.

5. Let d, n be a positive integers and let A∈Md(ℂ). Suppose An = 0. Determine the characteristic polynomial of A and prove that Ad = 0.

6. Let ζ = e2πi/13, a primitive 13 th root of unity. Prove that ℚ(ζ) contains exactly one subfield K such that [K : ℚ] = 6. Prove further that K is a Galois extension of ℚ and that K⊂ℝ.

7. Let k be a field, let d be a positive integer, and let Skd. Let f1, f2,... be a sequence of polynomials in k[x1,..., xd] with the property that for every sS, there exists a positive integer n such that fn(s)≠ 0. Prove that there exists a positive integer N such that for every sS, there exists nN such that fn(s)≠ 0.

8. Compute the character table of S3×ℤ/2ℤ.

Peter Linnell 2012-08-10