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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:
<!-- * Homework policies:
  * Late homework may be given directly to the grader, along with either  
  * Late homework may be given directly to the grader, along with either  
     (i) the instructor's permission, or (ii) a polite request for mercy.
     (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.
  * 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.
  * 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)
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, representation, linear and multilinear algebra, and the beginnings of ring theory.
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.


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


'''WEEK 1''':   
'''WEEK 1''':   
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Definition of group.  Associativity.  Inverse.   
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.   
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.
Homomorphisms.  The homomorphisms from F_k to G are in bijection with G^k.  Isomorphisms.
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'''WEEK 2''':
'''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.   
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.
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.
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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.
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.


==HOMEWORK 1 (due Sep 16)==
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.
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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c)  Explain why a) and b) imply that f must be identically 0.
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.  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




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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}.
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''.
<!--Below you will find a repository of homework problems.  Note that some of these problems are taken from Lang's ''Algebra''.
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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.)
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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