Algebra Prelim, Summer 2002

Instructions: do all problems.

  1. Let G be a finite group.
    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.

    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 $ \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).

    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

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

    Since Z0 $ \subseteq$ Z1 $ \subseteq$ Z2 $ \subseteq$ ..., 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