## Math 551 (Algebra I)

### Prof. Weibel

**Fall 2008**

**Homework Assignments**

All assignments are due the next class.

Assigned | Problems | |
---|---|---|

12/12/08 | In-class Final Exam | |

12/5/08 | Show that q(x,y)=dx²+(1-d)y² is equivalent to p(x,y)=x²+(1-d)dy² | |

12/2/08 | BA I, 6.1 #4: each T determines T' so that B(Tx,y)=B(x,T'y) | |

11/26/08 | Take-home exam (due Tuesday December 2) | |

11/21/08 | Find the matrix algebra decomposition of C[G]
when G is the dihedral group D_{7}.
| |

11/18/08 | BA I, 3.10 #5 (similar vs. equivalent matrices), 8 (nilpotent matrices), 10 (A²=A) | |

11/14/08 | BA I, 3.7 #7 (unimodular rows) and 3.8 #1 (cokernel of
Z^{2}→Z^{3})
| |

11/11/08 | BA I, 3.7 #1-2
(Normal form of matrices over Z and Q[λ])
| |

11/7/08 | Show that (m',s') ~(m",s") if (∃s)(m's"s=m"s's)
is an equivalence relation on M×S, that S ^{-1}M=M×S/~
is a well defined R-module, and that M→S^{-1}M is a homomorphism.
| |

11/4/08 | Show that every finite-dimensional M_{n}(F)-module
is semisimple (the ⊕ of simple modules)
| |

10/31/08 | Show that the coproduct of commutative algebras A and B is their tensor product A⊗B. | |

10/28/08 | Let 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 ^{-1}R is a DVR (see notes below).
| |

10/24/08 | BA I, 2.7 #10-11 (Chinese Remainder Theorem) | |

10/21/08 | Show 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 M_{n}(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).
| |

10/14/08 | Show that R[C_{3}]=R×C and
R[D_{3}]=R×R×M_{2}(R).
(R=real numbers, C=complex numbers)
| |

10/10/08 | First In-Class Exam
| |

10/3/08 | *Classify nonabelian groups of order p^{3},
p>2 BAI, 4.6 #9,11 (G is nilpotent iff it is a product of p-groups) | |

9/30/08 | Classify all non-abelian groups of order 12;
include D_{6}, A_{4}, and a metacyclic group.
| |

9/26/08 | Show that every group of order 18 is one of:
C_{18}, D_{9}, C_{6}×C_{3},
D_{3}×C_{3} or one other.
| |

9/23/08 | Classify all finite groups of order pq, where p,q are prime | |

9/19/08 | BAI, 1.11 #2(ed1) or #3(ed3)
(presentation of S_{n} by generators (1,i))*Show that the automorphisms A,B below induce a map from the Thomson group F to Aut([0,1]).
| |

9/16/08 | Show that every group of order n<15 (except 8,9,12) is cyclic or dihedral. | |

9/12/08 | BAI, 1.8 #4,11 (If |G|=2 mod 4, G has a normal subgroup of index 2.) | |

9/9/08 | Show 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. | |

9/5/08 | Determine the center of GL_{n}(F).
| |

9/2/08 | BAI, 1.2 #5,13; Classify all groups of orders 2 and 3 Show that GL _{2}(F_{2}) and D_{3} are isomorphic
| |

**For 9/19 HW:**
Richard Thompson's group *F* has presentation
< x_{0},x_{1},...|
x_{i}x_{n}x_{i}^{-1}=
x_{n+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^{-n}BA^{n} has slope 1 for t<1-2ε,
slope 1/2 for 1-2ε

**For 10/3 HW:**
If p>2, there are only two nonabelian groups of order p^{3}:
a metacyclic group (Z/p^{2}) ⋊ C_{p} and
an affine translation group (Z/p)^{2} ⋊ C_{p}.
If p=2, the only nonabelian groups of order 8 are D_{4} 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

**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 (p^{n}) for some n.
(This definition is from BA II, 9.2).

**For 11/11 HW:** a vector (r_{1},...,r_{n}) in
R^{n} is *unimodular* if the entries
r_{1},...,r_{n} generate R, i.e.,
(∃s_{i}) ∑ r_{i}s_{i}=1.

Return to syllabus or to Weibel's Home Page

Charles Weibel / weibel@math.rutgers.edu / November 10, 2008