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<!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--> | <!--(This argument shows that a solvable subgroup of A_n has size at most about (n-1)!, but this is far from sharp; e.g. I found a citation to a 1967 paper of Dixon which shows that a solvable subgroup of A_n has size at most k^n where k is around 2.88.)--> | ||
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8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A. | 8. Give an example of an abelian group A with a subgroup B such that B is not a direct summand of A. | ||
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An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--> | An amazing theorem of Margulis (superrigidity) shows that SL_2(Z) is very special among arithmetic groups in this way. For instance, there are only _finitely_ many isomorphism classes of n-dimensional representations of PSL_d(Z) for any d > 2; those groups are "rigid."--> | ||
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5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.) | 5. Let G be a finite group, let X be a set with G-action, and let chi be the character of the corresponding permutation representation. Show that, for each g in G, chi(g) is the number of elements of X fixed by g. (I am actually going to prove this in class.) | ||
Revision as of 02:46, 3 September 2013
Math 741
Algebra
Prof: Jordan Ellenberg
Grader: Evan Dummit
Ellenberg's office hours: Friday 3pm.
Grader's office hours: Monday 4pm. Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representation, linear and multilinear algebra, and the beginnings of ring theory.
SYLLABUS
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course.
WEEK 1:
Definition of group. Associativity. Inverse.
Examples of group: GL_n(R). GL_n(Z). Z/nZ. R. Z. R^*. The free group F_k on k generators.
Homomorphisms. The homomorphisms from F_k to G are in bijection with G^k. Isomorphisms.
WEEK 2:
The symmetric group (or permutation group) S_n on n letters. Cycle decomposition of a permutation. Order of a permutation. Thm: every element of a finite group has finite order.
Subgroups. Left and right cosets. Lagrange's Theorem. Cyclic groups. The order of an element of a finite group is a divisor of the order of the group.
The sign homomorphism S_n -> +-1.
WEEK 3
Normal subgroups. The quotient of a group by a normal subgroup. The first isomorphism theorem. Examples of S_n -> +-1 and S_4 -> S_3 with kernel V_4, the Klein 4-group.
Centralizers and centers. Abelian groups. The center of SL_n(R) is either 1 or +-1.
Groups with presentations. The infinite dihedral group <x,y | x^2 = 1, y^2 = 1>.
WEEK 4
More on groups with presentations.
Second and third isomorphism theorems.
Semidirect products.
WEEK 5
Group actions, orbits, and stabilizers.
Orbit-stabilizer theorem.
Cayley's theorem.
Cauchy's theorem.
WEEK 6
Applications of orbit-stabilizer theorem (p-groups have nontrivial center, first Sylow theorem.)
Classification of finite abelian groups and finitely generated abelian groups.
Composition series and the Jordan-Holder theorem (which we state but don't prove.)
The difference between knowing the composition factors and knowing the group (e.g. all p-groups of the same order have the same composition factors.)
WEEK 7
Simplicity of A_n.
Nilpotent groups (main example: the Heisenberg group)
Derived series and lower central series.
Category theory 101: Definition of category and functor. Some examples. A group is a groupoid with one object.
WEEK 8
Introduction to representation theory.
WEEK 10
Ring theory 101: Rings, ring homomorphisms, ideals, isomorphism theorems. Examples: fields, Z, the Hamilton quaternions, matrix rings, rings of polynomials and formal power series, quadratic integer rings, group rings. Integral domains. Maximal and prime ideals. The nilradical.
Module theory 101: Modules, module homomorphisms, submodules, isomorphism theorems. Noetherian modules and Zorn's lemma. Direct sums and direct products of arbitrary collections of modules.