Algebra Preliminary Exam, Spring 1987

Do any 8 problems

1. Let H be a subgroup of the finite group G, and let p be a prime. Prove that two distinct Sylow p-subgroups of H cannot lie in the same Sylow p-subgroup of G.
2. Let n be a positive integer and let R be a ring with a 1. Show that R has characteristic n if and only if R has a subring (with the same 1) isomorphic to (i.e. the integers modulo n).
1. Let A and B be normal subgroups of the group G.
1. Prove that is a normal subgroup of G.
2. Prove that is isomorphic to a subgroup of .
3. If G is finite and , prove that AB= G.
1. Let G be a simple group with a subgroup H of index 6. Prove that there exists a momomorphism .
2. Prove there are no simple groups of order 300.
2. Let R be the ring (so ). Define by for .
1. Show that for , .
2. Let . Show that is a unit if and only if .
3. Show that has no solution with .
4. Show that and are irreducible elements of R.
5. Deduce that R is not a UFD.
3. Let R be a UFD and let S be a multiplicatively closed subset of nonzero elements of R.
1. If u is irreducible in R, prove that u is either irreducible or a unit in S-1R.
2. Prove that S-1R is a UFD.
4. Let V be a vector space over and let be a linear transformation. Describe how V can be made into an -module via T.

Suppose V has basis (e1,e2,e3) and T is the linear transformation defined by T(e1) = 2e1, T(e2) = -4e2-4e3, and T(e3) = 9e2 + 8e3.

1. Express V as a direct sum of two nonzero -modules.
2. Calculate (x2 - 4x +4)e2.
3. If where f1 | f2 | ... | fn and f1 is not a unit, what are the possibilities for the ideals (fi)?
4. Express V as a direct sum of cyclic modules.
5. Does there exist such that ?
5. Let R be a PID, let p be a prime of R, and let M be the R-module where the ei and n are positive integers. Define and .
1. Prove that M(p) and pM are submodules of M.
2. Prove that .
3. In the case and p=x2+1, give an example of a finitely generated R-module M such that .
6. Let and , where is the ring of integers and . Prove that , where is the field of rational numbers.
7. List without repetition all the abelian groups of order 32 23. Which ones are cyclic?