Algebra Preliminary Exam, Fall 1987

Instructions: do all problems

1. Let G be a group with 56 elements. Prove that G is not simple.
2. Let G be a group and let . Prove that . Now suppose that G=HA where H and A are subgroups and A is abelian. Prove that there exists such that . Deduce that if G is nonabelian simple, then for all .
3. Let G be the group with the operation multiplication. Define by . Prove that is a group homomorphism, , and . Suppose for some positive integer . Is .
4. If H is a subgroup of the group G, let denote the normalizer of H in G. Suppose G is a finite group and P is a Sylow p-subgroup of G. Prove that .
5. Let R be a commutative ring. If I and J are ideals of R, define . Prove that (I:J) is an ideal of R.

Now suppose , , K = (I:(a)) and R/I is a domain. Prove that K=I and aK= I. Deduce that . Does the final assertion remain true if the hypothesis is dropped? ((a) denotes the ideal generated by a.)

6. Let K be a field. Prove that K[X] has infinitely many irreducible polynomials, no two of which are associates. (Consider p1p2 ... pn + 1). Suppose now , . Prove that there exists a homomorphism from K[X] to a domain with nonzero kernel such that .
7. Let R be a ring, let M be an R-module, and let be an R-module homomorphism. Prove that is a submodule of M.

Now suppose every submodule of M is finitely generated. Prove there exists an integer n such that . Deduce that if is onto, then is an isomorphism.

8. Let V be a vector space over and let be a linear transformation. Describe how V can be made into a -module via T.

Now let be a basis for V and suppose T(e1) = -e1 + 2e2, T(e2) = -2e1+3e2, T(e3) = -2e1+2e2 + e3. Find the Jordan canonical form for the matrix of T. Hence find the isomorphism type of V (as a -module) as a direct sum of primary cyclic modules. Does there exist a -module homomorphism of onto V?

Peter Linnell
Tue Apr 15 13:38:47 EDT 1997