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 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^mq 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.

2. Let K be a finite Galois extension of the rationals Q. Suppose that $\sqrt{2}$ and $\sqrt{3}$ 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 $\sqrt[8]{2}$$\notin$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 $\oplus$ B) $\otimes_{R}^{}$ C @ (A $\otimes_{R}^{}$ C) $\oplus$ (B $\otimes_{R}^{}$ C).

   (b)

   Let M be a finitely generated Z-module. Prove that if M $\otimes_{\mathbf {Z}}^{}$ 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 $\in$ M, set Ann(m) = {r $\in$ R | rm = 0}. Show there exists some w $\in$ 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
   Z_{j + 1} is the preimage in  G of Z(G/Z_j).
   Since Z₀ $\subseteq$ Z₁ $\subseteq$ Z₂ $\subseteq$ ..., 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.

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