Algebra Preliminary Exam, Fall 1987
Instructions: do all problems
- Let G be a group with 56 elements. Prove that G is not
- 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 .
- Let G be the group with the operation
multiplication. Define by .
Prove that is a group homomorphism, , and
. Suppose for some positive integer . Is .
- 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 .
- 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.)
- 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 .
- 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
- 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?
Tue Apr 15 13:38:47 EDT 1997