Math 551 homework (Fall 2008/Weibel)

## Math 551 (Algebra I)

### Prof. Weibel

Fall 2008
Homework Assignments
All assignments are due the next class.

Assigned Problems
12/12/08In-class Final Exam
12/5/08Show that q(x,y)=dx²+(1-d)y² is equivalent to p(x,y)=x²+(1-d)dy²
12/2/08BA I, 6.1 #4: each T determines T' so that B(Tx,y)=B(x,T'y)
11/26/08Take-home exam (due Tuesday December 2)
11/21/08Find the matrix algebra decomposition of C[G] when G is the dihedral group D7.
11/18/08BA I, 3.10 #5 (similar vs. equivalent matrices), 8 (nilpotent matrices), 10 (A²=A)
11/14/08BA I, 3.7 #7 (unimodular rows) and 3.8 #1 (cokernel of Z2Z3)
11/11/08BA I, 3.7 #1-2 (Normal form of matrices over Z and Q[λ])
11/7/08Show that (m',s') ~(m",s") if (∃s)(m's"s=m"s's) is an equivalence relation on M×S, that
S-1M=M×S/~ is a well defined R-module, and that M→S-1M is a homomorphism.
11/4/08Show that every finite-dimensional Mn(F)-module is semisimple (the ⊕ of simple modules)
10/31/08Show that the coproduct of commutative algebras A and B is their tensor product A⊗B.
10/28/08Let R be a UFD. (a) Show that R[x,y,...] is a UFD (use infinitely many variables)
(b) Let p be prime in R, and S=R-(pR). Show that S-1R is a DVR (see notes below).
10/24/08BA I, 2.7 #10-11 (Chinese Remainder Theorem)
10/21/08Show that every ideal of Z√(-5) is either principal or of the form zJ, J=(2,1-√(-5)), for some z.
10/17/08(a) Every right ideal of Mn(F) has dimension r n for some r, and
(b) the right ideals of dimension r n are in 1-1 correspondence with points of Grass(r,n).
10/14/08Show that R[C3]=R×C and R[D3]=R×R×M2(R). (R=real numbers, C=complex numbers)
10/10/08First In-Class Exam
10/3/08*Classify nonabelian groups of order p3, p>2
BAI, 4.6 #9,11 (G is nilpotent iff it is a product of p-groups)
9/30/08Classify all non-abelian groups of order 12; include D6, A4, and a metacyclic group.
9/26/08Show that every group of order 18 is one of: C18, D9, C6×C3, D3×C3 or one other.
9/23/08Classify all finite groups of order pq, where p,q are prime
9/19/08BAI, 1.11 #2(ed1) or #3(ed3) (presentation of Sn by generators (1,i))
*Show that the automorphisms A,B below induce a map from the Thomson group F to Aut([0,1]).
9/16/08Show that every group of order n<15 (except 8,9,12) is cyclic or dihedral.
9/12/08BAI, 1.8 #4,11 (If |G|=2 mod 4, G has a normal subgroup of index 2.)
9/9/08Show that the number of left cosets equals the number of right cosets;
If Hx=xH for all x, show that H is a normal subgroup of G.
9/5/08Determine the center of GLn(F).
9/2/08BAI, 1.2 #5,13; Classify all groups of orders 2 and 3
Show that GL2(F2) and D3 are isomorphic

For 9/19 HW: Richard Thompson's group F has presentation < x0,x1,...| xixnxi-1= xn+1 for every i<n. >
A is the piecewise linear automorphism of [0,1] whose graph has vertices (0,0), (.5,.25), (.75,.5), (1,1).
B is the PL automorphism of [0,1] whose graph has vertices (0,0), (.5,.5), (.75,.625), (.875,.75), (1,1).
A-nBAn has slope 1 for t<1-2ε, slope 1/2 for 1-2εn+2.)

For 10/3 HW: If p>2, there are only two nonabelian groups of order p3: a metacyclic group (Z/p2) ⋊ Cp and an affine translation group (Z/p)2 ⋊ Cp. If p=2, the only nonabelian groups of order 8 are D4 and Q.

For 10/17 HW: For any field F, the Grassmannian Grass(r,n) is the set of all r-dimensional subspaces of the vector space F. If r=1 it is the projective space of lines in n-space. If F is the field of real or complex numbers, this is a manifold (introduced by Hermann Grassmann). In general, Grass(r,n) has the structure of a smooth projective algebraic variety.

For 10/28 HW: A Discrete Valuation Ring (or DVR) is a Principal Ideal Ring with exactly one prime (up to units). This is equivalent to saying that every ideal is (pn) for some n. (This definition is from BA II, 9.2).

For 11/11 HW: a vector (r1,...,rn) in Rn is unimodular if the entries r1,...,rn generate R, i.e., (∃si) ∑ risi=1.