Algebra Prelim, Summer 2002

Instructions: do all problems.

1. 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 OQ denote the orbit containing Q with respect to this action. Prove that if H /\ NG(Q) = 1, then the orbit OQ has | H| subgroups in it.

(b)
Now suppose that | G| = pmq 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 NG(Q) = Q. Prove that G has a normal Sylow p-subgroup.

2. 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.

3. (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.

4. 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.

5. Let R be an integral domain. Prove that R is a field if and only if every R-module is projective.

6. 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 Z0 = {e} and

Zj + 1 is the preimage in  G of Z(G/Zj).

Since Z0 Z1 Z2 ..., there is a positive integer N such that ZN = ZN + 1 = ZN + 2.... Prove that ZN is equal to the intersection of all normal subgroups K in G such that Z(G/K) is the trivial group.

7. Explicitly find a simple (i.e. minimal) left ideal of the following ring of 2 X 2 matrices: M2(Q[x]/(x2 - 1)).

Peter Linnell
2002-08-17