**Algebra Preliminary Exam, Spring 1988**

Do all eight problems

- 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 .

- Find necessary and sufficient conditions on
- 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.

- Show that
- 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.

- Prove that an
- 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*R*/*I*-modules. - Show that is isomorphic to . You may use any results about tensor products you know.

- Show that is isomorphic to
- 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

Return to

Wed Jul 31 13:11:01 EDT 1996