GRADUATE ALGEBRA I FINALDUE 12/17/2010
There are 5 questions, each worth 20 points. Please justify all
answers.
1. The row and column indices in the n n Fourier matrix A run
from 0 to n 1, and the i, j entry isij , where = e2pii/n. This
matrix solves the following interpolation problem: Given complex
numbersb0, . . . , bn1, find a complex polynomial f(t) = c0 + c1 +
+ cn1tn1 such that f() = b .
(i) Explain how the matrix solves the problem.(ii) Prove that A
is symmetric and normal, and compute A2.(iii) Determine the
eigenvalues of A.
2. Let G be a finite group acting on a finite set S. For each g
G, let Sg = {s S | gs = s}, and let Gs bethe stabilizer of s.
(i) Prove Burnsides Formula:|G| (#orbits) =
gG
|Sg|.
(ii) There are 70 =(84
)ways to color the edges of an octagon, with four black and four
white. The group
D8 operates on this set of 70, and the orbits represent
equivalent colorings. Count the number ofequivalence classes.
3. (i) Prove the following generalization of the last Sylows
theorem: If |G| is divisible by pb, and H Ghas order pa with a b,
then the number of subgroups of G that both contain H and have
order pbis congruent to 1 modulo p.
(ii) Given a prime p, find an example of a finite group having
exactly 1 + p Sylow p-subgroups. Canthis be done for 1 + p? How
about for 1 + 3p?
4. (i) Let G = SL2(R). Using conjugation by elementary matrices,
show that every matrix A in G exceptfor I is conjugate to a matrix
having one of the forms(
0 11 d
)or(
0 11 d
).
(ii) Let A =(x yz w
)be a matrix in G and let t be its trace. Substituting t x for
w, the condition
that detA = 1 becomes x(t x) yz = 1. If we fix the trace t, then
the locus of solutions of thisequation is a quadric in x, y,
z-space. Describe the quadrics that arise this way, and decompose
theminto conjugacy classes.
5. (i) The conjugacy classes in Sn correspond to partitions of
n, and are determined by cycle structure.Which conjugacy classes of
Sn that are in An split into two conjugacy classes in An? Please
give anecessary and sufficient condition.
(ii) Compute the character table of the alternating group
A5.
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