MTH 623, Spring 2007
Brief Notes for Weeks 4-5
Monday, January 29, 2007 (5:00-6:15 PM)
We will finish up chapter 2, discussing the
correspondence theorem and some applications.
Here
is the lattice diagram of subgroups of S4.
Wednesday, January 31,
2007 (5:00-6:15 PM)
One more
application of the FHT – we prove that the group of inner automorphisms
of G is isomorphic to G/Z(G). (See the definition of center on p. 44.)
We begin
chapter 3, looking at the symmetric group and the more general issue of Ògroup
actionÓ.
We
define orbit, stabilizer, and prove some basic results about their
relationship.
Monday, February 5, 2007 (5:00-6:15 PM)
Note the
definition of conjugate on p.
43. (This is another ÒinvisibleÓ
idea, but critical in noncommutative environments such as linear algebraÉ) The conjugate of a permutation will be
a permutation with the same cycle structure.
Read to
page 48 and look at the following problems:
pages
45-46: 3.3, 6, 11,
page
48: 3.14, 15, 16.
Look at page 54: 3.27, 28, 29.
Note the
group of monomial matrices.
Wednesday, February 7,
2007 (No class – I am out of
town. We will make-up this missed
time in early April.)
Monday, February 12, 2007 (5:00-6:15 PM)
First, three
elementary remarks:
1. About the ÒalgorithmÓ defined by the
FHTÉ.
2. A normal subgroup is a union of
conjugacy classes.
3. On adding n copies of 1/nÉ.
Normal
subgroups of S5. (What
are they?)
We prove
the simplicity of A5. (n>3: Every even
permutation is a product of 3-cycles.
n>4: All 3-cycles are conjugates.)
We
prove the simplicity of A6. (If H is normal in A6, does it have a member which fixes 1?)
We
prove the simplicity of An. (If H is normal in An, n>6, then
H has a member which moves no more
than 6 elements of {1,É,n}.)
Wednesday, February 14,
2007 (5:00-6:15 PM)
We do
the orbit counting theorem and its applications.
Here
is the set of 3-colorings of the 2x2 square
(in preparation for the orbit-counting lemma.)
Last modified February 11, 2007