Instructions: do all eight problems
Notation: = integers, = rational numbers
Let G be a finite group.
Let A and B be subgroups of G such that and
AB = G. Prove that .
Let H be a subgroup of G such that [G:H] = 2 and let
. If m is the number of conjugates of h in G and n is
the number of conjugates of h in H, prove that either n=m or
n=m/2.
If the order of G is 105 and H is a subgroup of G of
order 35, prove that .
Let P be a nonnormal p-Sylow subgroup of the finite group
G. If is the normalizer of P in G, prove that
is nonnormal in G.
Let F_{1} and F_{2} be finite fields of orders q_{1} and q_{2}.
Prove that q_{i} is a power of a prime, say for i=1,2.
If , prove that p_{1}=p_{2} and that
is a divisor of .
Prove that x^{4}-2 is irreducible over .
Let K be the splitting field of x^{4}-2. Prove that the
Galois group of K over , , is of order
8.
Exhibit the correspondence (given by the Fundamental Theorem of
Galois theory) between the subgroups of and the
intermediate fields between and K.
State Nakayama's lemma.
Let R be a local commutative ring with maximal ideal M.
Let X be a finitely generated R-module. Show that if X/MX can
be generated by n elements, then so can X.
Construct an example of finitely generated
nonzero abelian groups A
and B so that .
If A and B are finitely generated abelian groups
such that and , prove that and .
Let R be a commutative ring and let A be an ideal of R.
Define the radical of A, denoted , by for some positive integer n}. You may assume
that is an ideal of R and that .
If P is a prime ideal of R such that , prove
that and as a consequence show that .
Prove that is the set of nilpotent elements of
R/A. (An element r is nilpotent if r^{n} = 0 for some positive
integer n.)