All assignments are due the next class.

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

12/9/10 | Show that k[G×H] is isomorphic to k[G]⊗k[H] as a k-algebra for all groups G, H. (The tensor product of algebras is always an algebra.) | |

12/6/10 | Set R=k[S_{3}].
If 1/6∈k, show that R is semisimple.
Identify R/J(R) if k has characteristic 2 or 3.
| |

12/2/10 | Show that the sum of two nilpotent ideals is nilpotent | |

11/29/10 | Show that every complex hermitian matrix (resp. real symmetric matrix) has a basis of eigenvectors. | |

11/14/10 | BA I, 6.1 #5 (the vector space of bilinear forms) | |

11/11/10 | BA I, 3.10 #2,10 (|coker(A)|=det(A); A²=A) | |

11/8/10 | BA I, 3.8 #1 (Abelian group with 3-generators, 2 relations) | |

11/4/10 | BA I, 3.7 #1-2
(Normal form of matrices over Z and Q[λ])
| |

11/1/10 | The left ideals of dimension n in M_{n}(k)
are in 1-1 correspondence with points of P^{n-1}
| |

10/28 | BA-I 2.16 #1,3 (Z is integrally closed, and
dim_{Q}Q(ζ_{p})=p-1.)
| |

10/21 | In-class Exam (group theory) | |

10/14 | Let H be the ring of holomorphic functions,
S={f | f(0)≠0}. Show that S^{-1}H is the ring of meromorphic
functions which are analytic at 0, and that this ring is noetherian.
| |

10/11 | (1) If F is a field, show that M_{n}(F) has no
nontrivial ideals. (2) Let I denote the kernel of the ring map
M_{n}(Z/p²) → M_{n}(Z/p) induced by
Z/p² → Z/p. Show that I²=0, i.e.,
show that AB=0 for every A,B in I.
| |

10/7 | Show that every complex representation of C_{n}
is a direct sum of eigenspaces V_{k}={v∈V|σv=exp(2πik/n)v}
(0≤k<n). Conclude that every irreducible representation of C _{n} is 1-dimensional.
| |

10/4 | If n>3 and G=GL_{n}(F), show that
[G,G]=SL_{n}(F).
Is SL_{n}(F_{p}) a simple group for n even?
(p prime>2).
| |

9/30 | (1) If every Sylow subgroup P_{i} in G is normal,
show that G=∏ P_{i}. (Cf. BAI, 4.6 #9,11)(2) If G is a finite nilpotent group, show that every Sylow subgroup is normal, and hence G=∏ P _{i}.
| |

9/27 | If |G|=12, show that G has a normal Sylow subgroup.
Using this, show that there are only 5 isomorphism classes of groups of
order 12, including D_{6}, A_{4}, and a metacyclic group.
| |

9/23 | Show that every group of order 6 is isomorphic to
either C_{6} or S_{3}; (Cf. BAI, 1.13 #3,4)If q<p and |G|=pq, show that G has a normal Sylow p-subgroup | |

9/20 | BAI, 1.12 #10; Show that the group of
monomial matrices in GL_{n}(F) is isomorphic to the wreath product G ι S _{n}
| |

9/16 | (1) Show than D_{n}=<r,t | r^n,t^2,rtrt>;
(2) If f:G→G' is onto, and H→G' is any group map,show that the kernel of G× _{G'}H→ H is isomorphic to ker(f)
| |

9/13 | BAI, 1.5 #5; 1.6 #2; 1.8 #11; 1.9 #1 | |

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

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Charles Weibel / weibel@math.rutgers.edu / October, 2010