Algebra Preliminary Exam, Fall 1988
Instructions: do all problems
- Prove that there are no simple groups of order 600.
- Let R be a principal ideal domain and assume that A, B, and
C are finitely generated R-modules. Suppose that is
isomorphic to . Prove that B is isomorphic to C.
- Prove that the Galois group of a splitting field K of an
irreducible polynomial p over the rational numbers acts
transitively on the roots of p. Show by examples that this theorem
does not necessarily hold if either
- K is not a splitting field, or
- p is reducible over .
- Prove that a group of order 255 is cyclic.
- Define what is meant by a solvable group. Prove that if , and H and G/H are solvable, then G is solvable.
- Let be a linear map where V is a finite
dimensional vector space over an algebraically closed field. Prove
that if 0 is the only eigenvalue of T, then Tn = 0 where .
- Prove that if is a homomorphism between
simple R-modules S and T, then either f is an isomorphism or
f is the zero homomorphism. (Recall that a nonzero R-module is
simple if 0 and the module itself are the only submodules.)
- Let R be a commutative ring with a 1.
- Prove that if M is a cyclic R-module, then M is isomorphic
to R/I for some ideal I of R.
- Prove that if M is a cyclic R-module and N is an
arbitrary R-module, then is isomorphic to N/IN
for some ideal I of R.
Wed Jul 31 13:18:36 EDT 1996