Algebra Preliminary Exam, Spring 1987
- 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.
- 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
- Let A and B be normal subgroups of the group G.
- Prove that is a normal subgroup of G.
- Prove that is isomorphic to a subgroup of .
- If G is finite and ,
prove that AB= G.
- Let G be a simple group with a subgroup H of index 6.
Prove that there exists a momomorphism .
- Prove there are no simple groups of order 300.
- Let R be the ring
(so ). Define
by for .
- Show that for , .
- Let . Show that is a unit if and only if
- Show that has no solution with .
- Show that and are
irreducible elements of R.
- Deduce that R is not a UFD.
- Let R be a UFD and let S be a multiplicatively closed
subset of nonzero elements of R.
- If u is irreducible in R, prove that u is either
irreducible or a unit in S-1R.
- Prove that S-1R is a UFD.
- 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.
- Express V as a direct sum of two nonzero -modules.
- Calculate (x2 - 4x +4)e2.
- If where f1 | f2 |
... | fn and f1 is not a unit, what are the possibilities for
the ideals (fi)?
- Express V as a direct sum of cyclic modules.
- Does there exist such that ?
- 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 .
- Prove that M(p) and pM are submodules of M.
- Prove that .
- In the case and p=x2+1, give an example of
a finitely generated R-module M such that .
- Let and ,
where is the ring of integers and . Prove that , where is the field of rational numbers.
- List without repetition all the abelian groups of order 32
23. Which ones are cyclic?
Wed Jul 31 14:01:23 EDT 1996