Fall 2013, TuTh, Science Center Hall A (NOTE NEW LOCATION)
My name: Andrew Cotton-Clay (please call me Andy)
Office: Science Center 527
Office Hours: M 12:30-1:30 and Th 2-3 or by appointment.
E-mail: acotton at math
Syllabus: Math 122 Syllabus
Course iSite: Math 122 iSite
1: due Sep 13
2: due Sep 20 selected solutions (2,4,6)
3: due Sep 27 (Erratum: Assume the
group is finite in 3b.) selected
solutions (1,4)
4: due Oct 4
Midterm 1 corrections: due Oct 11
5: due Oct 18
6: due Oct 25
7: due Nov 1
8: due Nov 8
No problem set due Nov 15 (just study for the midterm on Nov 13).
10: due Nov 22
(Note: Due to the
late posting of this problem set, if you e-mail me requesting an extension
to Monday, I promise to grant it.)
Sep 4: Introduction to groups (and semi-groups and monoids).
Associativity. Definition of subgroup. Some basic examples. See Artin
2.1-2.
Sep 6: Subgroups. Examples. Subgroups of the integers and greatest common
divisors. The symmetric group and cycle notation. Definition of group
homomorphism. See Artin 1.5 and 2.3.
Sep 9: Group homomorphisms. Examples, including the determinant and sign
homomorphisms. Images and Kernels. Normal subgroups. See Artin 1.5 and
2.5.
Sep 11: Isomorphisms. Automorphisms, conjugation, and the center of a
group. Left and right cosets. Normal subgroups and cosets. See Artin 2.6
and 2.8.
Sep 13: Quotients of groups by normal subgroups. Example of the group of
transformations of the real line of the form f(x) = ax+b. See Artin 2.8
and 2.12.
Sep 16: First isomorphism theorem and universal property of the quotient.
See Artin 2.12 (and 2.10); the universal property is included in Artin as
Proposition 7.10.13 (he leaves the proof to you). Direct products and
example of product of cyclic groups of relatively prime order. See Artin
2.11.
Also: Index of a subgroup, Lagrange's theorem, order of an element divides
the order of the group, Fermat's little theorem. See Artin 2.8.
Sep 18: Recap of some material from Sep 16, plus more examples of groups
and group extensions. One type, semidirect products, are not covered in
the book. See these
notes, attributed to Walter Neumann at Columbia, (only sections 1 and
2) for a nice discussion of these. We also discussed the quaternions; see
Artin 2.5.
Sep 20: We covered the extension {1,-1} -> Q -> K of the quaternions and
showed it was non-split. We also started our discussion of semidirect
products, seeing how to get one from a split extension, or equivalently
starting with a group G with subgroups H (required to be normal) and K
whose intersection is {1} and such that HK = G. See the notes linked for
Sep 18.
Sep 23: We showed the semidirect product is a group when defined from
scratch (see notes linked for Sep 18). We discussed examples of the
dihedral groups D_n, the group generated by translations and rotations
of the plane, and the group generated by translations, rotations, and
reflections as semidirect products (see Artin 6.3).
Sep 25: We defined rings, fields, and vector spaces. We covered subspaces,
linear maps, and quotients of vector spaces. We started to discuss bases.
See Artin 3.2 and 3.3 (rings are defined in Artin 11.1).
Sep 27: We covered bases and dimension, including the rank-nullity theorem
(Artin 3.4 and 4.1).
Sep 30: We covered matrices of linear transformations and change of basis.
(See Artin 3.5, 4.2, 4.3.) We defined conjugacy classes.
Oct 2: Midterm 1.
Oct 4: We covered eigenvectors, algebraically closed fields, and factoring
polynomials over algebraically closed fields. (See Artin 4.4 for
eigenvectors.)
Oct 7: We showed any endomorphism of a finite dimensional vector space
over an algebraically closed field has an eigenvalue. We discussed
invariant subspaces and the kernels of powers of T and the images of
powers of T as examples. (See Artin 4.4-5 for a different approach.)
Oct 9: We defined generalized eigenspaces and showed that, given an
endomorphism T of a finite dimensional vector space V over an
algebraically closed field, we have that V is a direct sum of the
generalized eigenspaces of T. (See Artin 4.6-7 for similar material from a
different approach.)
Oct 11: Jordan normal form, Artin 4.7, including discussion of a nice
block decomposition for a nilpotent operator.
--Caveat: Some of the following is approximate.--
Oct 16: Artin 5.1 and 6.2-3: orthogonal groups and identification of the
isometry group of R^n.
Oct 18: Finite subgroups of the isometry group of R^2. Beginning of
discussion of planar crystallographic groups. Artin 6.4-5.
Oct 21: Crystallographic groups in R^2. Group actions. Orbits and
stabilizers. The orbit-stabilizer theorem. Artin 6.6, 6.7-10.
Oct 23: Further discussion of group actions. Discussion of finite
subgroups of SO(3). Artin 6.12.
Oct 25: Identification of the tetrahedral group with A_4, the octahedral
group with S_4, and the icosahedral group with A_5. (Not explicitly in
Artin.)
Oct 28: Conjugacy classes and the class equation. Groups of
prime-power-order have center. The class equation for D_3. Artin 7.1-3.
Oct 30: Class equation for dihedral groups in general. Class equation for
the icosahedral group. Artin 7.2, 7.4.
Nov 1: Simplicity of the icosahedral group. Normalizers. Start of Sylow
theorem 1. Artin 7.4, 7.6, beginning of 7.7.
Nov 4: Sylow theorems 1, 2, and 3. Artin 7.7.
Nov 6: Classification of groups of order 21 and 12. Artin 7.7-8. Proofs
that groups of order p^k (k > 1) and pq cannot be simple. Study of simple
groups not of prime order, and elimination of all but 24, 36, 48, and 56
for those of size less than 60. (Not explicitly in Artin.)
Nov 8: Proof that a group of order 56 cannot be simple because there isn't
enough room for both sylows not to be normal. Proof that groups of order
24, 36, and 48 cannot be simple because in each case a non-normal sylow
subgroup would have small enough order that the action of the group by
conjugation on the sylow subgroups would give a nontrivial map to a small
symmetric group (and the kernel would then be normal). Proof that a simple
group of order 60 is isomorphic to A_5 by producing a nontrivial map to
S_5. (Not explicitly in Artin.)
Nov 11: Conjugacy classes and the class equation for the symmetric group.
Three cycles generate the alternating group. Sketch of simplicity of A_n
for n >= 5. Artin 7.5. Rubik's cube group: definition; proof that it is a
subset of the kernel of a map from ((Z/3)^8 x| S_8) x ((Z/2)^12 x| S_12)
to (Z/2) x (Z/2) x (Z/3) and brief discussion that it is the whole kernel.
(Not in Artin.)
Nov 13: Midterm 2.