03/01/10: Your regular instructor Professor Bobbe Cooper takes over
from now on. It has been a pleasure to work with you. Good luck!
03/01/10: Your regular instructor Professor Bobbe Cooper takes over
from now on. It has been a pleasure to work with you. Good luck!
03/01/10: Q: What is a nonsingular bilinear form (see HW 6)? A: The
same as a nondegenerate one.
02/19/10: I gave a correct indication on how to show that
Z(Sn) = {1} for n > 2, but totally misled you on showing
that Z(An) = {1} for n > 4 (In fact, it is also true for n
= 4, which may be shown directly, indeed.) A way to see that
Z(An) = {1} for n > 4 is to note that Z(An),
being the center of a normal subgroup, must also be normal in
Sn and use the (current homework) result on normal
subgroups of Sn. This is the only point at which we use
that result in the proof of simplicity of An.
02/12/10: Hint to Problem 4.3.30: Assuming the contrary, define
an action of Z2 on the conjugacy class.
02/06/10: An error in the textbook was pointed out to me by one of you
on Friday, but it is not on the official Errata list, see my previous
posting below. The error concerns Problem 4.1.10(a), which is on the
homework. The correct wording of Problem 4.1.10(a) is as
follows. "Prove that HxK is the union of left cosets xK in an orbit of
H acting by left multiplication on the set of left cosets of K." The
point is that it should not be assumed that any such orbit is
finite. This is a second place we have noticed in the text, in which
the authors seem to have finite groups in mind.
02/06/10: I have
found Errata
for Dummit and Foote's textbook.
02/05/10: When computing the size of the conjugacy class of
(12)(34)(56) in the symmetric group Sn at the end of the
class, I forgot the factor of 3! in the denominator, coming from the
group of permutations of the three factors in the above cycle
decomposition. Thus, |C((12)(34)(56))| = n!/(23 3!
(n-6)!). And then the argument went on saying that this number was
equal to n!/(2(n-2)!) only for n = 6.
02/03/10: I apologize about giving you a hint, which works only if G
is finite, to Problem 3.4.11. I have corrected the hint below and
discussed all related intricacies today in class.
02/01/10: Please, turn in your homeworks at the beginning of the
Monday class, when the homework is due, in the future.
01/28/10: Hint to Problem 3.4.11: If G were finite, you could
use the hint to Problem 8 in the text, i.e., consider a minimal
nontrivial normal subgroup A of G contained in H. To do this problem
in general, consider the derived series of H, see Section 6.1.
01/28/10: The second homework, due February 8, is posted. Good luck!
Remember, we will have a 30-minute quiz in the beginning of the class
on February 8.
01/20/10: The first homework, due February 1, is posted. Good luck!