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 tex2html_wrap_inline177. Prove that tex2html_wrap_inline179. Now suppose that G=HA where H and A are subgroups and A is abelian. Prove that there exists tex2html_wrap_inline189 such that tex2html_wrap_inline191. Deduce that if G is nonabelian simple, then tex2html_wrap_inline195 for all tex2html_wrap_inline197.
  3. Let G be the group tex2html_wrap_inline201 with the operation multiplication. Define tex2html_wrap_inline203 by tex2html_wrap_inline205. Prove that tex2html_wrap_inline207 is a group homomorphism, tex2html_wrap_inline209, and tex2html_wrap_inline211. Suppose tex2html_wrap_inline213 for some positive integer tex2html_wrap_inline215. Is tex2html_wrap_inline217.
  4. If H is a subgroup of the group G, let tex2html_wrap_inline223 denote the normalizer of H in G. Suppose G is a finite group and P is a Sylow p-subgroup of G. Prove that tex2html_wrap_inline237.
  5. Let R be a commutative ring. If I and J are ideals of R, define tex2html_wrap_inline247. Prove that (I:J) is an ideal of R.

    Now suppose tex2html_wrap_inline253, tex2html_wrap_inline255, K = (I:(a)) and R/I is a domain. Prove that K=I and aK= I. Deduce that tex2html_wrap_inline265. Does the final assertion remain true if the hypothesis tex2html_wrap_inline255 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 tex2html_wrap_inline279, tex2html_wrap_inline281. Prove that there exists a homomorphism tex2html_wrap_inline207 from K[X] to a domain with nonzero kernel such that tex2html_wrap_inline287.
  7. Let R be a ring, let M be an R-module, and let tex2html_wrap_inline295 be an R-module homomorphism. Prove that tex2html_wrap_inline299 is a submodule of M.

    Now suppose every submodule of M is finitely generated. Prove there exists an integer n such that tex2html_wrap_inline307. Deduce that if tex2html_wrap_inline207 is onto, then tex2html_wrap_inline207 is an isomorphism.

  8. Let V be a vector space over tex2html_wrap_inline315 and let tex2html_wrap_inline317 be a linear transformation. Describe how V can be made into a tex2html_wrap_inline321-module via T.

    Now let tex2html_wrap_inline325 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 tex2html_wrap_inline321-module) as a direct sum of primary cyclic modules. Does there exist a tex2html_wrap_inline321-module homomorphism of tex2html_wrap_inline321 onto V?

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