Algebra Preliminary Exam, Spring 1988
- Let G be a group and define by
- Find necessary and sufficient conditions on G such that
is a homomorphism.
- Under the conditions determined for (a), prove that is a normal subgroup of G and .
- Let be a group homomorphism with H an
abelian group. Suppose that N is a subgroup of G containing
. Prove that N is a normal subgroup of G.
- Let G be a group of order 99.
- Show that G is not a simple group.
- Show that G contains a subgroup of order 33.
- Prove that a finite abelian group is either cyclic or has at
least p elements of order p for some prime p.
- If S is a simple nonabelian group, prove that
contains a subgroup isomorphic to S. (Hint: consider conjugation.)
- Let R be a commutative ring with identity. A simple
R-module S is a module whose only submodules are 0 and S.
- Prove that an R-module S is simple if and only if there is a
maximal ideal such that S is isomorphic to
- Let R be a commutative ring with identity. Show that
simple R-modules exist.
- Let R be a ring with identity and let I be a (two-sided)
ideal in R. Let M and N be R-modules.
- Show that is isomorphic to M/IM as left
- Show that is isomorphic to . You may use any results about tensor products you
- Let E be an extension field of F with [E:F] = 11. Prove
that if with neither in F and if is an
F-automorphism of E, then
Wed Jul 31 13:11:01 EDT 1996