Do seven problems

- Let
*G*be a simple group of order 4312 = 2^{3}·7^{2}·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.

- Let
*R*be a UFD and let*I*_{1}⊆*I*_{2}⊆...⊆*R*be an ascending chain of principal ideals of*R*. Prove there exists a positive integer*N*such that*I*_{n}=*I*_{N}for all*n*≥*N*. - Let
*q*∈ℤ be a prime and let*P*be a nonzero projective ℤ-module (not necessarily finitely generated).- (a)
- Prove that
*P*≠*Pq*. - (b)
- Prove there exists a
ℤ-module epimorphism
*P*↠ℤ/*q*ℤ (you may assume that every subspace of a vector space has a direct complement). - (c)
- Prove that
*P*⊗_{ℤ}*P*≠ 0.

- 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*M*≌*N*. - Let
*d*,*n*be a positive integers and let*A*∈M_{d}(ℂ). Suppose*A*^{n}= 0. Determine the characteristic polynomial of*A*and prove that*A*^{d}= 0. - Let
ζ =
*e*^{2πi/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. - Let
*k*be a field, let*d*be a positive integer, and let*S*⊆*k*^{d}. Let*f*_{1},*f*_{2},... be a sequence of polynomials in*k*[*x*_{1},...,*x*_{d}] with the property that for every*s*∈*S*, there exists a positive integer*n*such that*f*_{n}(*s*)≠ 0. Prove that there exists a positive integer*N*such that for every*s*∈*S*, there exists*n*≤*N*such that*f*_{n}(*s*)≠ 0. - Compute the character table of
*S*_{3}×ℤ/2ℤ.

Peter Linnell 2012-08-10