(Assigned 2/2/00; due 2/9/00.)

- Find all finite multiplicative subgroups of the complex numbers.
(This means: finite sets of non-zero complex numbers that are closed
under multiplication and taking inverses.)
- Find some interesting finite multiplicative subgroups of the
quaternions.
- Suppose sigma is a map from the real numbers to the real numbers
that preserves all the field structure. (This means
sigma(

*x*+*y*) = sigma(*x*) + sigma(*y*), sigma(*x*)*y*) = sigma(*x*)sigma(*y*),sigma(1) = 1, sigma(0) = 0.)

Prove that sigma must be the identity map.

(Assigned 2/11/00; due 2/18/00.)

- List all the points and lines in the vector space
*k*^{2}, when*k*is the fieldZ /2Z . - Suppose
*A*is a two by two matrix with entries*a*,*b*,*c*, and*d*in a possibly non-commutative field*k*. Give precise algebraic conditions on*a*,*b*,*c*, and*d*for when*A*is invertible. (If*k*is commutative, you know the condition is that*a**d*-*b**c*is non-zero.) - Find an inclusion
*f*:H --> 2 by 2 complex matricesof the quaternions into 2 by 2 complex matrices, respecting addition and multiplication. (Hint: you might begin by finding three nice matrices

*I*,*J*, and*K*satisfying*I*^{2}=*J*^{2}=*K*^{2}= -(identity).You need to choose them also so that they multiply in the same way as

*i*,*j*, and*k*inH .

(Assigned 3/1/00; due 3/10/00.)

All of these problems concern a finite field *k* having
*q* = *p ^{r}* elements.

- An
*n*-dimensional vector space*V*over*k*has*q*elements. If^{n}*W*is an*m*-dimensional subspace of*V*, then the complement of*W*in*V*has*q*-^{n}*q*elements.^{m}a) Suppose that

*v*,...,_{1}*v*are linearly independent elements of_{m}*V*, and that*v*is another element of_{m+1}*V*. Show that*v*,...,_{1}*v*are linearly independent if and only if_{m+1}*v*does not belong to span(_{m+1}*v*,...,_{1}*v*). In particular, there are exactly_{m}*q*-^{n}*q*choices for^{m}*v*making_{m+1}*v*,...,_{1}*v*linearly independent._{m+1}b) Show that the number of ordered bases of

*V*is equal to(

*q*- 1)(^{n}*q*-^{n}*q*)(*q*-^{n}*q*)...(^{2}*q*-^{n}*q*).^{n-1}c) List all the ordered bases for the vector space

*k*^{2}, when*k*is the fieldZ /2Z . - Let
*G*=*GL*_{n}(*k*) be the group of invertible*n*by*n*matrices over*k*. This is a finite group (because there are only*q*to the power*n*^{2}*n*by*n*matrices over*k*). Prove that the largest power of*p*dividing the order of*G*is*q*^{n(n-1)/2}. - Find a group of invertible
*n*by*n*matrices over*k*having exactly*q*^{n(n-1)/2}elements.

(Assigned 4/7/00; due 4/19/00.)

All of these problems concern a finite field *k* having
*q* = *p ^{r}* elements.

- Suppose that
*V*is an*n*-dimensional vector space over*k*, and that*p*is an integer between 0 and*n*. Show that the number of*p*-dimensional subspaces of*V*is equal to(

*q*- 1)(^{n}*q*- 1)...(^{n-1}*q*- 1)^{n-p+1}/ (*q*- 1)...(^{p}*q*- 1).Here are hints for two different ways to solve this problem. For the first way, every

*p*-dimensional subspace*W*has a basis*v*,...,_{1}*v*. This gives an onto map from sets of_{p}*p*linearly independent vectors in*V*onto subspaces. You know how to count sets of linearly independent vectors. The number of sets mapping to one subspace*W*is equal to the number of bases of*W*; and you know that as well. For the second way, linear algebra tells you that the group*GL(V)*acts transitively on the collection of*p*-dimensional subspaces. Consequently the set of all*p*-dimensional subspaces is a homogeneous space*GL(V)/H*. Here the subgroup*H*is the stabilizer of any one particular*p*-dimensional subspace. The total number of subspaces is the order of*GL(V)*(which you know) divided by the order of*H*(which you can compute). - Calculate the limit of the formula in problem 1 as
*q*approaches 1. (Notice that the numerator and denominator approach 0, so it's a 0/0 indeterminate form.) - (This problem is quite difficult; the last part I don't know
how to do myself, so you shouldn't drive yourself crazy over it.)
a) Show that the formula in problem 1 gives a polynomial in

*q*with integer coefficients, of degree equal to*p(n-p)*.b) Show that the coefficients of the polynomial are actually

positive integers.c) The polynomial can be factored over the complex numbers into linear factors

*(q - r)*, where*r*is a root of unity. Describe these*p(n-p)*roots of unity as explicitly as you can. - Suppose
*V*is a symplectic vector space of dimension 2*n*. A subspace*L*of*V*is called*Lagrangian*if it has dimension*n*, and*X*.*Y*= 0 for all vectors*X*and*Y*in*L*. (Such subspaces exist: if*v*,...,_{1}*v*,_{n}*v'*,...,_{1}*v'*_{n}is a Darboux basis, then the span of

*v*,...,_{1}*v*is a Lagrangian subspace.) How many Lagrangian subspaces are there?_{n}