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'''Math 741'''
'''Math 741'''
Fall 2016


Algebra
Algebra
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Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]
Prof: [http://www.math.wisc.edu/~ellenber Jordan Ellenberg]


Grader:  Evan Dummit
Grader:  [https://www.math.wisc.edu/~eramos/ Eric Ramos].
<!-- * Homework policies:
* Late homework may be given directly to the grader, along with either
    (i) the instructor's permission, or (ii) a polite request for mercy.
* Assignments that are more than 1 page should be affixed in some reasonable way.
* Results from places (e.g., the internet) other than 741 and standard books must be cited.-->
 
Homework will be due on Wednesdays. 
 
JE's office hours:  Monday 12pm-1pm (right after class)
 
This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representations, linear and multilinear algebra, and the beginnings of ring theory.  A good understanding of the material of 741 and 742 are more than enough preparation for the qualifying exam in algebra.
 
==APPROXIMATE SYLLABUS==
 
This is a list of definitions and facts (not complete) with an estimate for when we'll encounter them in the course.
 
'''WEEK 1''': 
 
Definition of group.  Associativity.  Inverse. 
 
Examples of groups:  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.  Conjugacy classes of permutations. 
 
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'''


Ellenberg's office hours:  Tuesday 10am
Group actions, orbits, and stabilizers.


Grader's office hours: Wednesday 3pm
Orbit-stabilizer theorem.


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.
Cayley's theorem.


The main purpose of this website is to serve as a repository of homework problems.  Note that some of these problems are taken from Lang's ''Algebra''.
Cauchy's theorem.


HOMEWORK 1 (due Sep 20)
'''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.
 
==HOMEWORK 1 (due Sep 14)==


1.  Suppose that H_1 and H_2 are subgroups of a group G.  Prove that the intersection of H_1 and H_2 is a subgroup of G.
1.  Suppose that H_1 and H_2 are subgroups of a group G.  Prove that the intersection of H_1 and H_2 is a subgroup of G.
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b)  Prove that Q/Z is a ''divisible'' group:  that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x.  (Note that we write the operation in this group as addition rather than multiplication, which is why we write  ny for the n-fold product of y with itself rather than y^n)
b)  Prove that Q/Z is a ''divisible'' group:  that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x.  (Note that we write the operation in this group as addition rather than multiplication, which is why we write  ny for the n-fold product of y with itself rather than y^n)
c)  Prove that Q/Z is not finitely generated.  (Hint:  prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)
d)  Conclude that Q is not finitely generated.
4.  We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0.  Suppose f is a homomorphism from SL_2(Z) to Z.
a)  Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class.  Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).
b)  Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}.  (Recall that we say U1 and U2^{-1} are "conjugate".)  Explain why this also implies that f(U1) = -f(U2).
c)  Explain why a) and b) imply that f must be identically 0.
<!--5.  The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z.  Exhibit such a homomorphism.  Optional challenge problem:  exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z-->
5.  How many homomorphisms are there from the free group F_2 on two generators to S_3?  How many of these homomorphisms are surjective?
<!--
==HOMEWORK 1 (due Sep 14)==
1.  Suppose that H_1 and H_2 are subgroups of a group G.  Prove that the intersection of H_1 and H_2 is a subgroup of G.
2.  Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}.  List all the subgroups of S_3.
3.  We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer.  We denote the set of equivalence classes by Q/Z.  The operation of addition makes Q/Z into a group.
a)  For each n, prove that Q/Z has a subgroup of order n.
b)  Prove that Q/Z is a ''divisible'' group:  that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x.  (Note that we write the operation in this group as addition rather than multiplication, which is why we write  ny for the n-fold product of y with itself rather than y^n)
c)  Prove that Q/Z is not finitely generated.  (Hint:  prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)
d)  Conclude that Q is not finitely generated.
4.  We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0.  Suppose f is a homomorphism from SL_2(Z) to Z.
a)  Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class.  Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).
b)  Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}.  (Recall that we say U1 and U2^{-1} are "conjugate".)  Explain why this also implies that f(U1) = -f(U2).
c)  Explain why a) and b) imply that f must be identically 0.
5.  The argument above also shows that there is no nonzero homomorphism from SL_2(Z) to Z/pZ where p is a prime greater than 3. However, it leaves open the possibility that there is indeed a nonzero homomorphism from SL_2(Z) to Z/2Z.  Exhibit such a homomorphism.  Optional challenge problem:  exhibit a nonzero homomorphism from SL_2(Z) to Z/3Z
5.  Let G be the "affine linear group":  namely, G is the
==HOMEWORK 2 (due Sep 25)==
1.  Let V be the Klein 4-group in S_4.  Let Q be the symmetric group on the set DF of double-flips in S_4; there are 3 double flips, so Q is isomorphic to S_3.  If g is an element of S_4, we discussed in class that conjugation by g permutes the elements of DF.  So to each element of g, we have associated an element f(g) of Q.  More or less by definition, this defines a homomorphism f: G -> Q.  Show that this homomorphism is surjective and has kernel H, and thus that G/H is isomorphic to Q.
2.  Let F_2 be the free group on two generators, which we denote  x,y.  Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.
3.  Let H be a subgroup of G, and let N_G(H), the ''normalizer'' of H in G, be the set of elements g in G satisfying  g H g^{-1} = H.  Prove that N_G(H) is a subgroup of G.
4.  Let T be the subgroup of GL_2(R) consisting of diagonal matrices.  This is an example of a "Cartan subgroup," or "torus" (whence the notation T.)  Describe the normalizer N(T) of T in GL_2(R).  Prove that there is a homomorphism from N(T) to Z/2Z whose kernel is precisely T. 
5.  Prove that the group of inner automorphisms of a group is normal in the full automorphism group Aut(G).
6.  Let H and H' be subgroups of a group G, and let x be an element of G.  We denote by HxH' the set of elements of the form hxh', with h in H and h' in H', and we call HxH a ''double coset'' of the pair (H,H').
a)  Show that G decomposes as a disjoint union of double cosets of (H,H').
b)  Let G be GL_2(R) and let B be the subgroup of upper triangular matrices.  Prove that G decomposes into exactly two double cosets of (B,B).  (This is an example of the so-called *Bruhat decomposition* which is of great importance in the theory of algebraic groups and their representations.)
c)  Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1.  Describe the double coset decomposition of S_n into double cosets of (H,H).
==HOMEWORK 3 (due Oct 2)==
1.  Compute the centralizer of the triple flip (12)(34)(56) in S_6.  (Note:  this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)
2.  Warning:  a normal subgroup of a normal subgroup need not be normal!  Let H be the subgroup of S_4 generated by (12)(34).  Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4.  Show that H is normal in V_4, but H is not normal in S_4.
3.  Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.)  Prove that H_1 intersect H_2 also has finite index in G.
4.  Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4.  Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2.  This action can be thought of as a homomorphism from G to S_4.  Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.
5.  We proved two theorems in class:  Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p.  Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false.  Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.
6.  The group S_3 has only three conjugacy classes.  Prove that a finite group with at most three conjugacy classes has order at most 6.
7.  Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3.  Then S_n acts on X.  How many orbits are there?  (Hint:  the answer does not depend on n.)  OPTIONAL:  How many orbits are there on the space of ordered k-tuples, when n >= k? 
==HOMEWORK 4 (due Oct 9)==
1.  Recall that the commutator [x,y] is x y x^{-1} y^{-1}.  If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.
1a.  Show that G' is a normal subgroup of G.
1b.  Show that G/G' is an abelian group, which we call the abelianization, G^ab.
1c.  Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f.  Conclude that f factors through a homomorphism G/G' -> A.
1d.  Show that G' = G if and only if G has no nontrivial abelian quotient.  In this case, we say G is ''perfect''.
2.  Prove that any p-Sylow subgroup of GL_n(F_p) is contained in SL_n(F_p).
3.  Let U_n(F_p) be the subgroup of GL_n(F_p) consisting of upper triangular matrices with 1's on the diagonal.  Show that U_n(F_p) is a p-Sylow subgroup of GL_n(F_p).
4.  U_n(F_p) is called the *unipotent subgroup* of GL_n(F_p), because it has the property (not part of your homework, but easy to check) that every u in U_n(F_p) is unipotent, i.e. its characteristic polynomial is (x-1)^n.  It is a p-group (as you know by virtue of the last problem) so it must have a nontrivial center.  What is the center of U_nF_p)?
5.  U_3(F_p) is Shamgar Gurevich's favorite group, the *Heisenberg group" of order p^3.  Write down a composition series for U_3(F_p).
6.  Let A be the subgroup of Z^2 generated by the vector (1,0).  Let B be the subgroup of Z^2 generated by the vector (3,3).  Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.
7.  Suppose that G and H are groups and f: G -> H is a homomorphism.  Recall the definition of the abelianization G^ab from problem 1.
7a.  Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.
7b.  Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)
8.  Give two different composition series for S_4 and show that they have the same composition factors.
OPTIONAL:    We saw in class that a finitely generated abelian group that was torsion-free was in fact a free abelian group.  This totally dies without the hypothesis of abelianness.  To see this, give an example of a finitely generated group which is torsion free but which is not a free group, and which does not even *contain* any free group of rank greater than 1.
==HOMEWORK 5 (due Oct 16)==
1.  Let G be a group, thought of as a category with one object.  Show that a set with an action of G is the same thing as a functor from G to the category of sets.
2.  If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors.  We saw that the permutation representation of S_3 has a one-dimensional space of invariants.  Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then
dim V_X^G = number of orbits of X.
3.  (Invariants are functorial)  Let G be a group.  Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)).  The morphisms in this category are the ones described in class:  Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that
f(rho(g)(v)) = psi(g)(f(v))
for all g in G and all v in V.
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by
H_0((V) = V^G.
(In other words:  your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)
4.  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 V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables.  Describe the decomposition of V into irreducible representations of S_3.
6a.  Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.
6b.  As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.
6c.  The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G.  Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.
OPTIONAL:  (for people who know some topology.)  Let $X$ be a topological space and let $Paths/X$ be the category of paths up to isotopy as described in class.  Explain how to construct a bijection between {functors from Paths/X to the category FinSet of finite sets} and {finite covering spaces of X}.  (For simplicity you may assume X is path-connected.)
 
==HOMEWORK 6 (due Oct 23)==
1.  Let V_k be the space of homogenous degree-k polynomials in x_1, x_2, x_3.  Compute the character of S_3 acting on V_k, and, using this, give formulas for the multiplicity of the trivial rep, the sign rep, and the standard rep of S_3 in V_k.
2.  (Schur's lemma can be false over non-algebraically closed fields)  Let G be Z/3Z and let g be a generator of G.  Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.
2a.  Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible.  (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)
2b.  Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.
The following two problems involve the tensor product.  I am going to teach the general notion of tensor product of modules in this course, but I am expecting you already know what the tensor product of vector spaces.  If not, please review it by learning the definition and doing these problems, because having a familiarity with the vector space theory will be very helpful in learning the general story.
3.  Let V and W be complex vector spaces.  Let f be a linear transformation of V, and g a linear transformation of W. 
3a.  Show that there is a unique linear transformation F satisfying
F(v tensor w) = f(v) tensor g(w)
for all v in V and all w in W.  We denote this transformation by f tensor g.
3b.  Suppose V and W are finite dimensional.  Show that Trace(f tensor g) = Trace(f) Trace(g).  What does this say when f and g are both the identity transformation?
4.  Let V be a vector space.  We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form
v tensor w - w tensor v
for v,w in V.
Suppose dim V = n.  What is dim Sym^2 V?
5.  For any n, the group S_n has an (n-1)-dimensional representation, the "standard representation," on the space of vectors in C^n whose coordinates sum to 0.  When n=3, we proved in class that this representation was irreducible.  Prove that the standard representation is irreducible for all n.
6.  Give an example showing that the standard representation of S_n need NOT be irreducible over a field of characteristic p.
7.  Let H be a subgroup of G and let V be a representation of G.  Let chi_{G/H} be the character of the permutation representation of G on the set of cosets G/H, so that, by problem 1, chi_{G/H}(g) is the number of fixed points of g in its action on the cosets.  As discussed in class, the dimension of the G-invariant subspace V^G is given by the inner product of chi_V with the trivial character.  Prove that the dimension of V^H is given by the inner product of chi_V with chi_{G/H}.
==HOMEWORK 7 (due Oct 30)==
1.  When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space.  In this case, show that
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.
2.  Let G be the group of affine linear transformations of F_5; that is, it is the group of transformations x -> ax + b where a lies in (Z/5Z)^* and b lies in (Z/5Z).  Note that G has order 20.  Write H for the subgroup of G consisting of transformations fixing 0 (i.e. those for which b=0). 
We will work out all the irreducible representations of G.
2a.  First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H.  Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation.  (Hint:  it is probably easier to use the character of this representation than to prove irreducibility directly.) 
2b.  Now show that there are 4 1-dimensional representations of G.
2c.  Show that the irreducible representations of G you have constructed are the only ones.
3.  If G and H are finite groups, and V is a representation of G, and W is a representation of H, then V tensor W is a representation of GxH, and the character chi_{V tensor W} is given by
chi_{V tensor W}(g,h) = chi_V(g) chi_W(h).
3a.  Prove that V tensor W is irreducible if and only if V and W are irreducible.
3b.  Prove that all the irreducible representations of G x H are of the form V tensor W.
(Together, these propositions show that you can completely describe the representation theory of G x H in terms of that of G and that of H.)
4.  Suppose that V is an irreducible representation of S_n, and suppose that when we consider V as a representation of the alternating group A_n it is NOT irreducible.  Prove that V, considered as a representation of A_n, is the direct sum of two non-isomorphic irreducible representations.  Prove furthermore that chi_V(g) = 0 for all odd permutations g.  Give an example of such an irreducible representation of S_4.
OPTIONAL (because I couldn't quickly think of an easy way to do it!)  Prove that there exists such a representation of S_n for every n >= 3.
5.  (Fourier analysis over finite fields.)  I said in class that the representation theory that goes into Fourier analysis is different from the representation theory of finite groups we do in class, but that's not quite true; when you do Fourier theory over finite fields, the two theories come into much closer contact, with the bonus that we don't have to worry about issues of infinite sums that I hand-waved away in class.
Let F be the field of p elements and let V be the space of complex-valued functions on F; so V is a p-dimensional space.  Let G be the group Z/pZ (in other words, it is the additive group of the field.)
Now V is a representation of G, in the same way we discussed in class:  if f is a function in V, and a is an element of g, then the function gf is defined by
gf(x) = f(x+a)
5a.  Describe the breakup of V into irreducible representations of G, which are all 1-dimensional (not just what the characters are, explicitly decribe the irreducible constituents of V as subspaces of V!)
5b.  There is a natural norm on V which sends f to ||f|| = sum_x |f(x)|^2.
Given any f, we have a unique decomposition f = sum_i f_i, where f_i lies in the irreducible constituent V_i of V.
Prove that ||f|| = sum_i ||f_i||.
OPTIONAL (for people who are taking analysis) Explain why ||f_i|| is correctly thought of as a "Fourier coefficient" and why the fact proved in 5b is the finite-field analogue of Parseval's identity.
==HOMEWORK 8 (due Nov 6)==
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.
1a. Show that any nonzero Boolean ring has characteristic 2.  (In other words, show that a+a=0 for all a in R.)
1b. Show that every Boolean ring is commutative.
2. Let phi : R -> S be a homomorphism of commutative rings (recalling that rings for us have a multiplicative identity 1 and that ring homomorphisms take 1 to 1.)
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.
3. Let Nil(R) denote the set of nilpotent elements of a ring R; that is, the set of x such that x^k = 0 for some positive integer k.
3a. Show that Nil(R) is an ideal whenever R is a commutative ring.  Assume R is commutative for the remainder of this exercise.
3b. Describe Nil(Z/720Z).
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.
3d. Show that Nil(M_2(R)) is not an ideal of the matrix algebra M_2(R). (That is, it is neither a left nor a right ideal.)
4.  We say a ring is Noetherian if any ascending chain of ideals I_1 < I_2 < I_3 < …. eventually stabilizes.  This is a somewhat funny-looking condition if you haven't encountered it before, but it turns out to be a very useful way of formalizing the notion that a ring is "not too bad."
4a.  Show that Z is Noetherian.
4b.  Let R be the ring of continuous functions from the real numbers to the real numbers.  Show that R is not Noetherian.
5.  '''How algebraists do calculus'''.  The ''ring of dual numbers'' is the ring A = C[e]/(e^2).  We think of e as an infinitesimal number; that is, something so small that its square is 0.  This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows.  Let f be a polynomial in C[x].  Show that there is a unique polynomial g in C[x] such that
f(x+e) - f(x) = eg(x)
and that in fact g is the derivative of f.  In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!
6.  Write down all the ideals of C[e]/e^2.  (There are three.)
7.  Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.
== HOMEWORK 9 (Due Nov 13) ==
1. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.
2. It follows from 1. that if S is a commutative ring and J is a maximal ideal of S, that J/J^2 is a vector space over the field S/J.  Compute the dimension of J/J^2 as a vector space over S/J when
2a.  S = Z, J = (5);
2b.  S = C[x,y], J = (x,y)
2c.  Suppose S = C[x,y]/(f) and J = (x,y).  Then the dimension of J/J^2 depends on the choice of the polynomial f (where, to ensure that J/(f) makes sense, you should assume that f is in J).  Compute the possible values of this dimension and give an example of an f realizing each possibility.  (NOTE:  this is secretly, or not-so-secretly, another instance of differential geometry as carried out by algebraists...)
3.  An ''idempotent'' element of a ring is an element e satisfying e^2 = e.
3a.  If e is an idempotent, show that 1-e is also an idempotent.
3b.  Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R.  Show that eR and (1-e)R are both two-sided ideals.  Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)
3c.  Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism.  Finally show that the ring homomorphism
f_{e} x f_{1-e}: R -> eR x (1-e)R 
is actually an isomorphism of rings.  In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.
4.  Let k be an algebraically closed field, and let D be a finite-dimensional division algebra over k.  Prove that D = k.  (Hint:  if alpha is an element of D which is not in k, prove that there is some polynomial f(x) such that f(alpha) = 0.  Then think back to our proof in class that the quaternion algebra with complex coefficients was not a division algebra.)
5.  The ''center'' of a ring R is the subring of elements commuting with every element of R.
5a.  Show that the center of M_n(Q) is the ring of scalar matrices.
5b.  Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.  Compute the central idempotents of C[S_3].
6.  '''How algebraists do differential equations.'''  We define the ''Weyl algebra'' W of differential operators in one variable as follows.  Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C.  Differentiation, which we denote d, is also a linear transformation on V.  So we let W be the subring of End(V) generated by x and d.  We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.
6a.  Describe the submodule of V which is annihilated by the element d-x in W.
6b.  Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = g'.  (This is another way algebraists do calculus!) 
6c. Use the facts from 6b. to show that the center of the Weyl algebra is just the field of constants. (Remark:  you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)
7.  If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M.  Show that Ann(M) is a two-sided ideal. 
8.  Let M be a left module for R, and let m be an element of M.  We denote by Ann(m) the set of elements r in R such that rm = 0.
8a.  Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).
8c.  Let V,W be as in question 6.  Show that Ann(e^x) is the left ideal W(d-1).
==HOMEWORK 10 (due Nov 22)==
1.  Let f: R -> S be a homomorphism of rings.  Then S naturally has the structure of R-bimodule by left and right multiplication (technically, r acts by multiplication by f(r).)  If M is a left R-module, then S tensor_R M is a left S-module.  On the other hand, if N is an S-module, then N can also be considered as an R-module by pullback of the action.  These two operations are functors:  the first, a functor from left R-modules to left S-modules, the second, a functor from left S-modules to left R-modules.
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module.  On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.
2.  Prove a fact I stated in class:  if A and B are abelian groups (i.e. Z-modules) which are finite, and |A| and |B| are relatively prime, then A tensor_Z B = 0.
3.  Prove the converse:  if A and B are finite abelian groups and A tensor_Z B = 0, then |A| and |B| are relatively prime.
4.  Let R be a commutative ring with 1.  If R has a unique maximal ideal M, R is called a "local ring".
4a. If R is a local ring, show that every element of R that is not in M is a unit.  (I think Rob H. proved this in class.)
4b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.
5a.  If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs.  Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module.  In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.
5b.  Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.
5c.  Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.
5d.  Optional:  Give an example of a ring (with 1) which is *not* isomorphic to its opposite ring.  (This is not so easy, I think!  I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)
6.  Let f: C[x] tensor_C C[y] -> C[z] be the unique map satisfying f(x^a tensor y^b) = z^{a+b}.  Show that f(m tensor n) is nonzero for any nonzero pure tensor m tensor n, but f is not injective.  (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)
7.  Let R be a commutative local ring with maximal ideal m, and A -> B a map of R-modules. 
7a.  Show that if A -> B is surjective, than so is A tensor R/m -> B tensor R/m.  (In fact, this has nothing to do with R being local.)
7b.  Show that if B is a finitely generated R-module, the CONVERSE is true:  if A tensor R/m -> B tensor R/m is surjective, then so is A -> B (use Nakayama)
7c.  Show that the converse is not true without the hypothesis of finite generation.  For instance, let R = C[[t]], and give an example of a map A -> B of R-modules which is not surjective, but such that A/tA -> B/tA is surjective.
<!--Below you will find a repository of homework problems.  Note that some of these problems are taken from Lang's ''Algebra''.
==HOMEWORK 1 (due Sep 20)==
1.  Suppose that H_1 and H_2 are subgroups of a group G.  Prove that the intersection of H_1 and H_2 is a subgroup of G.
2.  Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}.  List all the subgroups of S_3.
3.  We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer.  We denote the set of equivalence classes by Q/Z.  The operation of addition makes Q/Z into a group.
a)  For each n, prove that Q/Z has a subgroup of order n.
b)  Prove that Q/Z is a ''divisible'' group:  that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x.  (Note that we write the operation in this group as addition rather than multiplication, which is why we write  ny for the n-fold product of y with itself rather than y^n)
c)  Prove that Q/Z is not finitely generated.  (Hint:  prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)


4.  Let F_2 be the free group on two generators x,y.  Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.
4.  Let F_2 be the free group on two generators x,y.  Prove that there is a automorphism of F_2 which sends x to xyx and y to xy, and prove that this automorphism is unique.
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c)  Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1.  Describe the double coset decomposition of S_n into double cosets of (H,H).
c)  Let G be the symmetric group S_n, and let H be the subgroup of permutations which fix 1.  Describe the double coset decomposition of S_n into double cosets of (H,H).
==HOMEWORK 2 (due Sep 27)==
1.  Prove that the symmetric group S_n is generated by transpositions.  Hint:  proceed by induction on n.  Let H be the subgroup of S_n consisting of permutations fixing 1.  Observe that H is isomorphic to S_{n-1}, so that H is generated by transpositions.  Explain why it suffices to show that, for any g in S_n, the coset gH contains a transposition.  Then explain why this is the case.
2.  Given that the symmetric group S_n is generated by transpositions, show that there are no nontrivial homomorphisms from S_n to Z/pZ for any odd prime p.
3.  Compute the centralizer of the triple flip (12)(34)(56) in S_6.  (Note:  this is the kind of problem that a computer algebra package like sage is very good at, and I encourage you to learn to use sage well enough to check your answer.)
4.  Warning:  a normal subgroup of a normal subgroup need not be normal!  Let H be the subgroup of S_4 generated by (12)(34).  Then H is a subgroup of the Klein 4-group V_4, which we have already shown is normal in S_4.  Show that H is normal in V_4, but H is not normal in S_4.
5.  The '''quaternion group''' Q is a non-abelian finite group of order 8.  Its elements are 1, -1, i, -i, j, -j, k, -k; it satisfies g^2 = -1 for all g except +-1, and ijk = 1.  (Note that the notation "-i" means "the product of i with -1" and -1 is understood to be central in Q.)  Prove that every subgroup of Q is normal.
6.  (finishing example done in class)  Let Gamma be the group F<x,y> / (x^2 = 1, y^2 = 1).  (Remember, this means the quotient of the free group by the group generated by all conjugates of x^2 and y^2.)  Prove that Gamma is infinite.  Prove that Gamma has a subgroup R generated by xy isomorphic to Z, that R is normal, and that the quotient Gamma/R is isomorphic to Z/2Z. 
7.  Suppose a group G has the property that every element in g satisfies g^2 = 1.  Show that G must be abelian.
==HOMEWORK 3 (due Oct 4)==
1.  Multiplication by -1 is an automorphism of Z/nZ.  Let a be the homomorphism Z/2Z -> Aut(Z/nZ) which sends the nontrivial element of Z/2Z to multiplication by -1.  Then we have a semidirect product of Z/pZ by Z/2Z as defined in class.  This group of order 2n is called a ''dihedral group.'' and is denoted D_n (or sometimes D_{2n}).
1a.  Compute the center of D_n.  (Note that the answer depends on n!)
1b.  Show that all involutions of D_n are conjugate to each other if and only if n is odd.
2.  (More on direct products that we didn't do in class.)  Suppose that G is a semidirect product of N with H.  Suppose furthermore that H is also normal in G.  Prove that G is a direct product of N with H.
3.  Let Q be the quaternion group from last week's homework.  Show that Q does not decompose as a semidirect product N \rtimes H, apart form the trivial cases N = {e} and N = Q.
4.  The ''affine linear group'' of degree n is the group of transformations from R^n to R^n of the form x -> Ax + b, where A is a matrix in GL_n(R) and b is a vector in R^n.  Prove that the affine linear group is a semidirect product R^n \rtimes GL_n.
5.  The ''ordinary triangle group''  T(p,q,r) is the group with presentation <x,y | x^p = y^q = (xy)^r = 1>.  This is a very interesting family of groups which arises naturally in many contexts in number theory and topology.
6a.  Prove that T(2,2,n) is isomorphic to the dihedral group D_n.
6b.  Prove that T(2,3,3) is finite and compute its order.  Is it isomorphic to a finite group we've discussed in class?
(It is a beautiful fact, but outside the scope of this course, that T(p,q,r) is finite if and only if 1/p + 1/q + 1/r > 1.)
7.  Let G be a subgroup of GL_n(R) which contains SL_n(R).  Prove that G is a normal subgroup of GL_n(R).
8.  Let G be a group and let H_1 and H_2 be subgroups of G of finite index (that is, the number of cosets in G/H_i is finite, i = 1,2.)  Prove that H_1 intersect H_2 also has finite index in G.
9.  Recall that the commutator [x,y] is x y x^{-1} y^{-1}.  If G is a group, the commutator subgroup of G, denoted G' or [G,G], is the subgroup of G generated by all commutators [x,y] for x,y in G.
9a.  Show that G' is a normal subgroup of G.
9b.  Show that G/G' is an abelian group.
9c.  Show that if f: G -> A is a homomorphism, with A abelian, then G' is contained in ker f.  Conclude that f factors through a homomorphism G/G' -> A.
9d.  Show that G' = G if and only if G has no nontrivial abelian quotient.  In this case, we say G is ''perfect''.
10.  Under what conditions on (p,q,r) is the triangle group T(p,q,r) perfect?
==HOMEWORK 4 (due Oct 16)==
1.  Let X be a set with a transitive action of a group G, which we think of as a homomorphism f: G -> Sym(X).  Let H_x be the stabilizer of an element x of X.
1a.  If x' is another element of X, show that H_{x"} and H_x are conjugate subgroups.  (Hint:  this is not true without the hypothesis of transitivity, so make sure to use it!)
1b.  Show that the kernel of f is the intersection of all the conjugates of H_x in G.  (This subgroup is called the ''normal core'' of H_x.)
2.  This week, a certified mathematical grown-up asked on MathOverflow what the index-3 subgroups of SL_2(Z/pZ) were.  Let's do it ourselves for homework.  Let p be a prime, and suppose H is an index-3 subgroup of SL_2(Z/pZ).  I do not claim that the proof sketched below is necessarily the easiest, but it allows us to cover some ground that we won't get to do in lecture.
2a.  Using the coset action, show that the normal core of H is the kernel of a homomorphism f from SL_2(Z/pZ) to S_3.
2b.  A ''unipotent matrix'' is a matrix A such that (A-I)^n = 0 for some positive integer n.  Prove that a unipotent element of SL_2(Z/pZ) has order dividing p.
2c.  You may use the fact that unipotent matrices generate SL_2(Z/pZ).  Given this fact, show that f must be trivial when p > 3.
2d.  Using the above, show that SL_2(Z/pZ) has no index-3 subgroups when p > 3.
2e.  (extra credit) What can you say about the minimal index of a proper subgroup of SL_2(Z/pZ)?
3.  Let H be a subgroup of G of index 2.  Prove that H is normal.
4.  We proved two theorems in class:  Lagrange's theorem, which says that the order of a subgroup of G divides |G|, and Cauchy's theorem, which says that if p divides |G| then there exists a subgroup of G of order p.  Cauchy's theorem is a kind of partial converse to Lagrange's theorem, but the full converse is false.  Give an example of a finite group G and a divisor n of |G| such that you can prove there is no subgroup of G of order n.
5.  Let X be the set of lines in F_3^2 (here a line means a 1-dimensional linear subspace) so that |X| = 4.  Let G be the projective linear group PGL_2(F_3), so that G acts on X via its action on F_3^2.  This action can be thought of as a homomorphism from G to S_4.  Show that this homomorphism is an ISOMORPHISM from PGL_2(F_3) to S_4.
6.  The group S_3 has only three conjugacy classes.  Prove that a finite group with at most three conjugacy classes has order at most 6.
7.  Let X be the set of ordered triples of elements of {1,..,n}, for some n >= 3.  Then S_n acts on X.  How many orbits are there?  (Hint:  the answer does not depend on n.)
8.  Suppose that G and H are groups and f: G -> H is a homomorphism.  Recall the definition of the abelianization G^ab from the previous problem set.
8a.  Show that the composition G -> H -> H^ab factors through a unique homomorphism G^ab -> H^ab, which we denote f^ab.
8b.  Show that if f: G -> H and g: H -> Q are homomorphisms, then f^ab o g^ab = (f o g)^ab.
(For those reading MacLane, this constitutes a proof that abelianization is a '''functor''' from the category of groups to the category of abelian groups.)
==HOMEWORK 5 (due Oct 23)==
1.  Show that there are only two different isomorphism classes of groups of order 14.  (Hint:  prove that such a group must be a semidirect product of a group of order 7 by a group of order 2.)
2.  (variant of Burnside's Lemma)  Let G be a finite group and H a normal subgroup.  Suppose G acts on a finite set X.
2a.  Show that if x and y lie in the same H-orbit, and g is an element of G, then gx and gy also lie in the same H-orbit.  Thus, G/H has a natural action on the set of H-orbits X/H.
2b.  Show that the average number of fixed points in X of an element of the coset gH is the same as the number of fixed points of the element gH of G/H in its action on X/H.
3.  A ''central extension'' of a group G by an abelian group A is a group E, together with a surjective homomorphism E -> G whose kernel is central in E and is isomorphic to A.
3a.  Show that the quaternion group is a central extension of (Z/2Z)^2 by Z/2Z.
3b.  Give an example of another nonabelian group which is a central extension of (Z/2Z)^2 by Z/2Z and which is not isomorphic to the quaternion group.
4.  Prove that a central extension of an abelian group is nilpotent.
5.  Give two different composition series for S_4 and show that they have the same composition factors.
6.  Suppose that A is a finitely generated abelian group, and B is a subgroup of A.
6a. Prove that B is finitely generated and that rank(B) <= rank(A).
6b. Prove that rank(A/B) = rank(A) - rank(B).
7.  For this problem, you may use the fact (which I should have asked on homework earlier) that if H_1 and H_2 are subgroups of G, then
[G: H_1 intersect H_2] <= [G:H_1][G:H_2].
Prove that if G is a simple group, d is an integer greater than 1, and S is a subgroup of index d in G, then |G| <= d^d.
Conclude that, in particular, A_5 has no subgroup of index 3.  On the other hand, exhibit a subgroup of A_5 of index 5.
(Note:  d^d is not a sharp bound; can you get a bound of d factorial instead?)
<!--(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.
9.  Let A be the subgroup of Z^2 generated by the vector (1,0).  Let B be the subgroup of Z^2 generated by the vector (3,3).  Show that A and B are isomorphic to each other, but Z^2 / A is not isomorphic to Z^2 / B.
==HOMEWORK 6 (due Oct 30)==
1.  In this problem, you will construct a functor F from the category of finite sets to the category of complex vector spaces.  I'll tell you what it does on objects:  if X is a finite set, then F(X) is the vector space spanned by a set of basis elements {e_x}_{x in X}, so that dim F(X) = |X|.  Your job:  complete the definition of F by describing the map from Mor_{Finite Sets}(X,Y) to Mor_{Vector Spaces}(F(X),F(Y)), and showing that your map respects composition.
2.  If (V, rho) is a representation of a group G, denote by V^G the space of vectors in V such that gv = v for all g in G; these are called _invariant_ vectors.  We saw that the permutation representation of S_3 has a one-dimensional space of invariants.  Prove that if X is a finite set with G-action, and V_X the corresponding permutation representation of G, then
dim V_X^G = number of orbits of X.
3.  (Invariants are functorial)  Let G be a group.  Then there is a category called C[G]-Mod whose objects are complex representations of G (i.e. pairs (V,rho) where V is a complex vector space and rho is a homomorphism from G to GL(V)).  The morphisms in this category are the ones described in class:  Mor((V,rho),(W,psi)) is the set of linear maps f from V to W such that
f(rho(g)(v)) = psi(g)(f(v))
for all g in G and all v in V.
Show that there is a functor H_0 from the category of representations of G to the category of vector spaces whose action on objects is given by
H_0((V) = V^G.
(In other words:  your job again is to explain how to associate to a morphism from (V,rho) to (W,phi) a morphism from V^G to W^G, in a way that's compatible with composition.)
<!--4.  The group PSL_2(Z) has a presentation of the form [U,V|U^2 = V^3 = 1].  Thus, an n-dimensional representation of PSL_2(Z) is precisely a pair of n x n matrices (rho(U),rho(V)), such that rho(U)^2 = rho(V)^3 = 1.  Prove that for every n > 1 there are uncountably many isomorphism classes of n-dimensional representations of PSL_2(Z).  (Extra credit:  show that there are uncountably many isomorphism classes of _irreducible_ representations.)
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."-->
<!--
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.)
6.  Let V be the space of homogeneous degree-3 polynomials in variables x_1, x_2, x_3; then S_3 acts on V by permutation of the three variables.  Describe the decomposition of V into irreducible representations of S_3.
7a.  Show that if chi is the character of a representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi.
7b.  As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.
7c.  The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G.  Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.
==HOMEWORK 7 (due Nov 6)==
1.  Let G be a finite group, let X be a finite 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.)
2a.  Show that if chi is the character of a finite-dimensional representation of a finite group G, then chi(g^{-1}) is the complex conjugate of chi(g) for all g.
2b.  As a consequence, show that if g is conjugate to g^{-1} for all g in G, then chi(g) is real.
2c.  The groups D_p and S_n have the property that g is conjugate to g^{-1} for _every_ g in G, so that every character is a real-valued function on G.  Give an example of a group G where this is not the case, and give an example of a representation of G whose character takes non-real values.
3.  Let V and W be complex vector spaces.  Let f be a linear transformation of V, and g a linear transformation of W. 
3a.  Show that there is a unique linear transformation F satisfying
F(v tensor w) = f(v) tensor g(w)
for all v in V and all w in W.  We denote this transformation by f tensor g.
3b.  Suppose V and W are finite dimensional.  Show that Trace(f tensor g) = Trace(f) Trace(g).  What does this say when f and g are both the identity transformation?
4.  Let V be a vector space.  We define Sym^2 V to be the quotient of V tensor V by the subspace generated by all elements of the form
v tensor w - w tensor v
for v,w in V.
Suppose dim V = n.  What is dim Sym^2 V?
5.  Show that Z/pZ tensor Z/qZ is zero when p and q are distinct primes.
==HOMEWORK 8 (due Nov 13)==
1.  (Schur's lemma can be false over non-algebraically closed fields)  Let G be Z/3Z and let g be a generator of G.  Then G acts by cyclic permutations on the set {1,2,3}; let P be the corresponding permutation representation on the REAL (not complex!) vector space generated by e_1,e_2,e_3.
1a.  Show that P breaks up as the direct sum of the trivial representation and a 2-dimensional representation V which is irreducible.  (Remember, we are not working over the complex numbers, so you cannot use the criterion for irreducibility in terms of the character that we worked out in class for the complex case; you need to prove directly that V has no nontrivial proper subrepresentation.)
1b.  Show that the map from V to V induced by g is G-equivariant, but is neither zero nor a scalar multiplication.
2.  If V_1 and V_2 are representations (not necessarily irreducible!) of G, show that <chi_V_1, chi V_2> is a non-negative integer.
3.  When a formula always yields a non-negative integer, it is often because it is secretly a formula for either the cardinality of some finite set or the dimension of some finite-dimensional vector space.  In this case, show that
<chi_{V_1}, chi_{V_2}> = dim_C Hom_G(V_1,V_2)
where we recall that Hom_G(V_1,G_2) refers to the space of G-equivariant homomorphisms from V_1 to V_2.
4.  If A is an abelian group, show that every irreducible representation of A is 1-dimensional.  (Hint:  show that there is a vector which is an eigenvector for every element of A!)
5.  Let G be the group of order 20 which is the semidirect product of A by H, where A is Z/5Z and H is (Z/5Z)^* with its natural action on A.  We will work out all the irreducible representations of A.
5a.  First of all, consider the 5-dimensional permutation representation of G acting by left multiplication on the cosets of H.  Show that this representation decomposes as a direct sum of the trivial representation and an irreducible 4-dimensional representation.  (Hint:  it is probably easier to use the character of this representation than to prove irreducibility directly.) 
5b.  Now show that there are 4 1-dimensional representations of G which factor through the quotient G/A (i.e. such that rho(A) is the identity.)
5c.  Using the decomposition of the regular representation, show that the irreducible representations of G you have constructed are the only ones.
==HOMEWORK 9 (due Nov 20)==
0. Find all ring homomorphisms from Z to Z/30Z. [Warning: there are 8 of them, not 30.]
1. If R is a ring (not necessarily with 1) such that a^2 = a for every a in R, R is called a Boolean ring.
1a. Show that any nonzero Boolean ring has characteristic 2.  (In other words, show that a+a=0 for all a in R.)
1b. Show that every Boolean ring is commutative.
2. Let phi : R -> S be a homomorphism of commutative rings with 1.
2a. If I is an ideal of S, show that phi^(-1) (I) is an ideal of R.
2b. If P is a prime ideal of S, show that phi^(-1) (P) is either R or a prime ideal of R.
2c. If M is a maximal ideal of S and phi is surjective, show that phi^(-1) (M) is a maximal ideal of R.
2d. Find a homomorphism phi : R -> S and a maximal ideal M of S such that phi^(-1) (M) is not a maximal ideal of R.
3. Let Nil(A) denote the set of nilpotent elements of a ring A, and let R be a commutative ring.  In class it was observed that Nil(R) is an ideal of R.
3a. Find (a generator of) Nil(Z/720Z).
3b. Show that Nil(R/Nil(R)) = 0.
3c. If x is in Nil(R) and u is a unit, show that u+x is a unit.
3d. Show that Nil(S), where S is the 2x2 matrices with real coefficients, is not an ideal of S.
4a. If M is an R-module and I is an ideal of R such that im = 0 for all i in I and m in M, show that M carries the structure of an R/I-module.
4b. If S is a commutative ring with 1 and J is a maximal ideal of S, show that J/J^2 is a vector space over S/J.
5. A "divisible element" x of a Z-module is one such that for each positive integer k, there exists a y such that ky=x. Show that a free Z-module cannot contain a nonzero divisible element.
6. Prove that the direct sum of any collection of free R-modules is also a free R-module.
7. In this problem you will prove that the module M given by the direct product of copies of Z, indexed by the positive integers, is not a free Z-module. Therefore, assume that M is a free Z-module with basis B, and let N be the direct sum of copies of Z, as a submodule of M.
7a. Show there is a countable subset B1 of B such that N is contained in the submodule N1 generated by B1. [Hint: N is countable.]
7b. Show that M/N1 is a free Z-module.
7c. Let S = {(b1, b2, b3, ...) : b_i = i! or -i!}. Show that there is some s in S which is not in N1. [Hint: S is uncountable.]
7d. For any s in S but not N1, show that for each positive integer k, there exists some m in M with s-km in N1.  Conclude that s+N1 is a divisible element of M/N1.
7e. Use problem 5 to obtain a contradiction.
8. In class I said "rank" is not always well-defined for R-modules if R is not commutative.  Here is an example demonstrating this assertion. Let M be the countably infinite direct product of copies of Z as in exercise 7, and let R be the endomorphism ring of M.  Define phi_1(a1,a2,a3,a4,...) = (a1,a3,a5,...) and phi_2(a1,a2,a3,a4,...) = (a2,a4,a6,...).
8a. Show that {phi_1, phi_2} is a free basis for R, as a left R-module. [Hint: let psi_1(a1,a2,a3,...) = (a1,0,a2,0,...) and psi_2(a1,a2,a3,...) = (0,a1,0,a2,...). Compute the products of the phi_i and psi_j in either order, and use the result to show that the phi_i are a basis.]
8b. Show that R is isomorphic to R^n for every positive integer n.
==HOMEWORK 10 (due Dec 4)==
1.  '''How algebraists do calculus'''.  The ''ring of dual numbers'' is the ring R = C[e]/(e^2).  We think of e as an infinitesimal number; that is, something so small that its square is 0.  This algebraic notion of infinitesimal is the basis for the algebraic notion of differentiation, which works as follows.  Let f be a polynomial in R[x].  Show that there is a unique polynomial g in C[x] such that
f(x+e) - f(x) = eg(x)
and that in fact g is the derivative of f.  In other words, the usual definition of "derivative" works just fine in this context, without any use of the notion of limit!
2.  Write down all the ideals of C[e]/e^2.  (There are three.)
3.  Show that any right ideal of the ring M_2(Q) of 2x2 matrices is an ideal of the form I_V described in class for some subspace V of Q^2; that is, it consists precisely of those linear transformations whose image is contained in V.
4.  An ''idempotent'' element of a ring is an element e satisfying e^2 = e.
4a.  If e is an idempotent, show that 1-e is also an idempotent.
4b.  Suppose R has a unit (we are actually always supposing this, but I just want to remind you of it for this problem) and suppose e is a ''central'' idempotent, i.e. an idempotent which commutes with all elements of R.  Show that eR and (1-e)R are both two-sided ideals.  Moreover, show that eR and (1-e)R are also subrings of R (with the warning that the identity in eR is not 1, which is not contained in eR, but rather e.)
4c.  Show that the map f_e: R -> eR defined by f_e(r) = er is a ring homomorphism.  Finally show that the ring homomorphism
f_{e} x f_{1-e}: R -> eR x (1-e)R 
is actually an isomorphism of rings.  In other words, the presence of a central idempotent induces a decomposition a ring as a direct product.
5.  The ''center'' of a ring R is the subring of elements commuting with every element of R.
5a.  Show that the center of M_n(Q) is the ring of scalar matrices.
5b.  Show that the center of the group ring C[G] is the set of elements sum f(g) g such that f(g) is a class function.
6.  '''How algebraists do differential equations.'''  We define the ''Weyl algebra'' W of differential operators in one variable as follows.  Multiplication by x is a linear transformation on the complex vector space V of all smooth functions on C.  Differentiation, which we denote d, is also a linear transformation on V.  So we let W be the subring of End(V) generated by x and d.  We note that x and d satisfy the relation xd = dx - 1; in particular, the Weyl algebra is not commutative.
6a.  Describe the submodule of V which is annihilated by the element d-x in W.
6b.  Show that, for any polynomial f(x), fd-df = -f', and also show that for any polynomial g(d), gx-xg = -g'.  (This is another way algebraists do calculus!) 
6c. Use the facts from (b) to show that the center of the Weyl algebra is just the field of constants. (Remark:  you will use in the course of this proof the fact that a polynomial whose derivative is 0 is constant -- this would not be true if we were working in positive characteristic, as the example of x^p considered as a polynomial over F_p demonstrates.)
7.  If M is a left module for a ring R, we denote by Ann(M) the set of elements r in R such that rm = 0 for all m in M.  Show that Ann(M) is a two-sided ideal. 
8.  Let M be a left module for R, and let m be an element of M.  We denote by Ann(m) the set of elements r in R such that rm = 0.
8a.  Show that Ann(m) is a left ideal, and show that the submodule Rm of M generated by m is isomorphic to the left module R/Ann(m).
8c.  Let V,W be as in question 6.  Show that Ann(e^x) is the left ideal W(d-1).
==HOMEWORK 11 (due Dec 13)==
1.  Let f: R -> S be a homomorphism of rings.  As discussed in class, if M is a left R-module, then S tensor_R M is a left S-module, and if N is an S-module, then N can also be considered as an R-module by pullback of the action.
Prove that if M is a finitely generated left R-module, then S tensor_R M is a finitely generated left S-module.  On the other hand, give an example to show that N can be finitely generated as a left S-module but not finitely generated when considered as a left R-module.
2.  Recall from class that when V and W are complex vector spaces,  W^dual tensor V is naturally isomorphic to the space of linear homomorphisms Hom(W,V).
Now suppose that V and W are left C[G]-modules, i.e. complex representations of a group G.  Show that W^dual tensor_C[G} V is the maximal G-invariant quotient of Hom(W,V); in other words, the quotient of Hom(W,V) by the subspace generated by all functions F of the form F(w) =  f(gw) - gf(w) for some function f in Hom(W,V) and some g in G.
3.  '''How algebraists do logic.'''  Why are Boolean rings called Boolean rings?  Because they model some features of the Boolean arithmetic that governs propositional logic.  Let A be a collection of propositions.  We define two operations on A as follows.  By a + b we mean the negation of a xor b, while by ab we mean "a or b."
3a.  It is not a priori obvious that these operations satisfy the ring axioms, but assume this for the moment.  Show that a^2 = a for all a (so A is a boolean ring) and 2a = 0 for all a (so A has characteristic 2.)
3b.  Suppose we have a consistent assignment of truth-values to the propositions in A.  Show that the set of true statements of A forms an ideal of A.
3c.  (Optional for people who like propositional logic) Show that a(b+c) = ab + ac with the given operations.
4.  Let K be Q[x]/(x^3-2).
4a.  Show that K is a field.
4b.  Show that K tensor_Q K decomposes as the direct sum of two fields.  What are they? [Note: K tensor_Q K is a ring!]
5.  There was some discussion in class about when we can "go back and forth" between right modules and left modules, as is the case with group rings.
5a.  If R is a ring, we denote R^op ("the opposite ring") to be the ring whose elements are those of R, but whose multiplication law x_opp is defined by s x_opp r = rs.  Show that if M is a left R-module, then the map M x R^op -> M sending (x,r) to rx makes M into a right R^op module.  In particular, if there is a RING ISOMORPHISM from R to R^op, then every left R-module can be thought of as a right R-module.
5b.  Verify that the map sending g to g^{-1} extends by linearity to give a ring isomorphism between C[G] and C[G]^op.
5c.  Construct a ring isomorphism between M_n(Q) and M_n(Q)^op.
5d.  Optional:  Give an example of a ring (with 1) which is not isomorphic to its opposite ring.  (This is not so easy, I think!  I found a MathOverflow thread with several examples, none of which came with a short proof of the non-isomorphism -- try your hand!)
6. . Let R be a commutative ring with 1.  If R has a unique maximal ideal M, R is called a "local ring".
6a. If R is a local ring, show that every element of R that is not in M is a unit.
6b. Conversely, if R is such that the set of nonunits forms an ideal I, show that R is a local ring with maximal ideal I.
7.  The inclusion of S_3 into S_4 associated to the inclusion of {1,2,3} into {1,2,3,4} induces a ring homomorphism from C[S_3} to C{S_4].  Let V be a representation of C{S_3].  Then, as discussed in class,  V tensor_C[S_3] C[S_4] is a representation of S_4, called the '''induced representation''' of V.
7a.  Show that dim_C (V tensor_C[S_3] C[S_4]) = 4 dim_C V.
7b.  When V is the trivial representation of S_3, decompose (V tensor_C[S_3] C[S_4]) as a sum of irreducible representations of S_4.
-->
<!--==HOMEWORK PROBLEMS TO GIVE IN FUTURE SEMESTERS==
Let f: C[x] tensor_C C[y] -> C[z] be the map of rings with f(x) = f(y) = z.  Show that f(m tensor n) is not 0 for any pure tensor m tensor n, but f is not injective.  (This is a warning not to try to check injectivity by checking that no pure tensors are killed by the map.)
Put in some problems about the affine linear group (say, that the normalizer of the non-normal subgroup is itself)
-->

Latest revision as of 21:36, 6 September 2016

Math 741

Fall 2016

Algebra

Prof: Jordan Ellenberg

Grader: Eric Ramos.

Homework will be due on Wednesdays.

JE's office hours: Monday 12pm-1pm (right after class)

This course, the first semester of the introductory graduate sequence in algebra, will cover the basic theory of groups, group actions, representations, linear and multilinear algebra, and the beginnings of ring theory. A good understanding of the material of 741 and 742 are more than enough preparation for the qualifying exam in algebra.

APPROXIMATE SYLLABUS

This is a list of definitions and facts (not complete) with an estimate for when we'll encounter them in the course.

WEEK 1:

Definition of group. Associativity. Inverse.

Examples of groups: 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. Conjugacy classes of permutations.

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.

HOMEWORK 1 (due Sep 14)

1. Suppose that H_1 and H_2 are subgroups of a group G. Prove that the intersection of H_1 and H_2 is a subgroup of G.

2. Recall that S_3 (the symmetric group) is the group of permutations of the set {1..3}. List all the subgroups of S_3.

3. We can define an equivalence relation on rational numbers by declaring two rational numbers to be equal whenever they differ by an integer. We denote the set of equivalence classes by Q/Z. The operation of addition makes Q/Z into a group.

a) For each n, prove that Q/Z has a subgroup of order n.

b) Prove that Q/Z is a divisible group: that is, if x is an element of Q/Z and n is an integer, there exists an element y of Q/Z such that ny = x. (Note that we write the operation in this group as addition rather than multiplication, which is why we write ny for the n-fold product of y with itself rather than y^n)

c) Prove that Q/Z is not finitely generated. (Hint: prove that if x_1, .. x_d is a finite subset of Q/Z, the subgroup of Q/Z generated by x_1, ... x_d is finite.)

d) Conclude that Q is not finitely generated.

4. We will prove that there is no homomorphism from SL_2(Z) to Z except the one which sends all of SL_2(Z) to 0. Suppose f is a homomorphism from SL_2(Z) to Z.

a) Let U1 be the upper triangular matrix with 1's on the diagonal and a 1 in the upper right hand corner, as in class, and let U2 be the transpose of U1, also as in class. Show that (U1 U2^{-1})^6 = identity (JING 1, TAO 0) and explain why this implies that f(U1) = f(U2).

b) Show that there is a matrix A in SL_2(Z) such that A U1 A^{-1} = U2^{-1}. (Recall that we say U1 and U2^{-1} are "conjugate".) Explain why this also implies that f(U1) = -f(U2).

c) Explain why a) and b) imply that f must be identically 0.


5. How many homomorphisms are there from the free group F_2 on two generators to S_3? How many of these homomorphisms are surjective?




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