Algebra Preliminary Exam, Spring 1981
- Prove that a group of order pn, where p is a prime and , has a nontrivial center.
- Let G be a group of order pq, where p and q are primes
and p < q. Prove that G has only one subgroup of
Show that if H is a subgroup of order 12 in a group G of
order 36, then H is a normal subgroup of G. (Map G to the
automorphisms of the set of right cosets of H.)
NOTE: there are
of groups of order 36 with subgroups of order
12 which are not normal.
- Describe all abelian groups of order 36, up to
- Prove that if p(x) is a polynomial irreducible over a field
F, and if and are roots of p(x) in some
extension field of F, then and are
isomorphic. What happens if p(x) is reducible?
- Let L be a normal extension of a field F, let
, and let H be a subgroup of G. Prove that
- if for all , then K
is a subfield of L;
- if K is normal over F, then H is a normal subgroup of
- Let R be a commutative principal ideal domain (PID) with a 1.
If , show that a and b have a greatest common divisor (i.e. d divides a and b, and if g divides a and b,
then g divides d).
- Show that is not a PID.
- Let R be a commutative ring with identity, and let I,J be
ideals in R. Let for
- Show that IJ is an ideal of R.
- Show that if I+J = R, where , then (recall that R has a 1).
- Suppose further that R is a domain, and that
for all ideals I,J in R. Prove that R is a field. (Take
Wed Jul 31 21:30:22 EDT 1996