Algebra Prelim, Summer 2002
Instructions: do all problems.
- Let G be a finite group.
- (a)
- Let H and Q be subgroups of G. Note that H
acts on the set of conjugates of Q via conjugation. Let
O_{Q} denote the orbit containing Q with respect to this
action. Prove that if
H /\ N_{G}(Q) = 1, then the orbit
O_{Q} has | H| subgroups in it.
- (b)
- Now suppose that
| G| = p^{m}q where p and q are distinct primes
and m is a positive integer. Let Q be a Sylow q-subgroup of
G and suppose that
N_{G}(Q) = Q. Prove that G has a normal
Sylow p-subgroup.
- Let K be a finite Galois extension of the rationals
Q.
Suppose that and are both elements of K.
Show that
Gal(K/Q) has a normal subgroup N of index
4. Show further that if | N| is odd, then
K.
- (a)
- Let R be a ring. Let A and B be right R-modules and let C
be a left R-module. Prove that
(A B) C @ (A C) (B C).
- (b)
- Let M be a finitely generated
Z-module. Prove that if
M M = 0, then M = 0.
- Let R be a commutative Noetherian ring with unity and let M be a
nonzero R-module. Given m M, set
Ann(m) = {r R | rm = 0}. Show there exists some w M such that
Ann(w) is
a prime ideal of R.
- Let R be an integral domain. Prove that R is a field if and only
if every R-module is projective.
- Denote the center of a group by
Z(·). Let G be a
finite group with identity element e. Define a sequence of
subgroups of G inductively by
Z_{0} = {e} and
Z_{j + 1} is the preimage in G of Z(G/Z_{j}).
Since
Z_{0} Z_{1} Z_{2} ..., there is a positive integer
N such that
Z_{N} = Z_{N + 1} = Z_{N + 2}.... Prove that
Z_{N} is equal to the intersection of all normal subgroups K
in G such that
Z(G/K) is the trivial group.
- Explicitly find a simple (i.e. minimal) left ideal of the following
ring of
2 X 2 matrices:
M_{2}(Q[x]/(x^{2} - 1)).
Peter Linnell
2002-08-17