Due date | Reading | Written assignment |
---|---|---|

Wednesday, August 23 | Skim chapter 0 to make sure you're comfortable with it. Then read Chapter 1.1. Memorize the definition of a group, and be comfortable with the content of the theorems and examples. | Begin homework 1 |

Friday, August 25 | Chapter 1.2: Know what the dihedral group of order 2n is and why it has 2n elements. Be comfortable thinking of its elements both as operations on a regular polygon and as "words" in the letters r and s. | Post your problem to D2L |

Monday, August 28 | Chapter 1.3: Know what the symmetric groups are and how many elements they have. Be able to go back and forth between "cycle decompositions" and more intuitive notation for permutations. (That is, be comfortable through about midway through page 31.) | Post your referee report to D2L |

Wednesday, August 30 | Chapter 1.3: Be very comfortable working with cycle decompositions. That is, be skilled at multiplying, inverting, or computing the order of permutations presented as cycle decompositions. | Write up homework 1 |

Friday, September 1 | Chapter 1.4: Understand the
definition of the general linear group, GL_{n}(R),
and know that we can substitute other nice objects like Q
and C for R. This may require you to go back over
your linear algebra notes about what makes a matrix invertible.
If so, read that too.
| Post your problem to D2L |

Monday, September 4 | Labor day: cast aside the yoke of oppression. | Post your referee report to D2L. (This deadline is fluid, because Labor day.) |

Wednesday, September 6 | Chapter 1.5: Know the quaternion group. Then reread chapter 1.3. | Write up homework 2 |

Friday, September 8 | Chapter 1.6: Understand that a homomorphism is a function that respects the group structure, and know what it means for two groups to be isomorphic. | Post your problem to D2L |

Monday, September 11 | Chapter 1.7: Know the definition of a group action, and be comfortable with the first four examples. | Post your referee report to D2L |

Wednesday, September 13 | Chapter 1.7: Be very comfortable with the prose discussion of group actions, and with all five examples. | Write up homework 3 |

Friday, September 15 | Chapter 2.1: Know the definition of a subgroup, and be comfortable with the examples and non-examples. Know the proof of the "subgroup criterion", but also understand the the criterion is almost useless. | Post your problem to D2L |

Monday, September 18 | Chapter 2.2: Be very comfortable with the stabilizer and kernel subgroups coming from a group action. Recognize centralizer and normalizer subgroups as a special case of these. | Post your referee report to D2L |

Wednesday, September 20 | Chapter 2.2: Be comfortable with all the examples of centralizers and normalizers. Know the definition of the center of a group, recognize its importance, and determine the centers of all our standard examples. | Write up homework 4 |

Friday, September 22 | Chapter 2.3: Understand that "cyclic" means "generated by one element", and recognize that you understand cyclic groups (and cyclic subgroups of larger groups) very well. In particular, know exactly what all the subgroups of a cyclic group are, and under what circumstances one subgroup contains another. | Post your problem to D2L |

Monday, September 25 | Chapter 2.4: Understand how to describe a subgroup with a list of generators, and recognize cyclic subgroups as a special case of this. Take the discussion beginning with "When G is non-abelian the situation is much more complicated." seriously. | Post your referee report to D2L |

Wednesday, September 27 | Chapter 2.5: Make sure you're comfortable reading these lattices. | |

Friday, September 29 | Chapter 3.1: Know what cosets are, and understand that they partition the big group. Be comfortable with the examples through page 80. | Post your problem to D2L |

Monday, October 2 | Chapter 3.1: Understand the idea behind multiplication of cosets, and that it only works if the subgroup is normal. Be comfortable with the definition of a quotient group, and with the various equivalent characterizations of normality. | Post your referee report to D2L |

Wednesday, October 4 | Chapter 3.2: Internalize the statements of Lagrange's theorem and Cauchy's theorem. Be comfortable with the content of Propositions 14 and 15 about when HK is a subgroup. | Write up homework 6 |

Friday, October 6 | Chapter 3.3: Be very comfortable with the content of all four isomorphism theorems. (It's not important to know which is which.) | Post your problem to D2L |

Monday, October 9 | Chapter 3.4: Know what simple groups and composition series are, and understand why we care about them. | Post your referee report to D2L |

Wednesday, October 11 | Chapter 3.5: Know what
transpositions are, and know that S_{n} is generated by
its transpositions. Understand that a homomorphism, called
"sign", from S_{n} to {1,-1}, with the property that the
sign of every transposition is -1, exists. Be able to compute the
sign of any permutation, using its cycle decomposition and this
property. (Do not worry too much about the definition of the sign
map. We will discuss it in class on Wednesday or Friday.)
| Write up homework 7 |

Friday, October 13 | Chapter 3.5: Know at least one
definition of the sign homomorphism, and the definition of the
alternating group A_{n}. Understand that A_{n} is
normal in S_{n}, and know its role in composition series
for S_{n}.
| Post your problem to D2L |

Monday, October 16 | Chapter 4.1: Recall what actions
are, and be clear on the relationship between actions of G on a
set A and homomorphisms from G to the symmetric group S_{A}.
| Post your problem to D2L |

Wednesday, October 18 | Chapter 4.2: Recognize that every group acts on itself, on all its quotients, and on the coset space of any subgroup, by left multiplication. Understand that this creates homomorphisms to the appropriate symmetric groups. Understand the number-theoretic cleverness involved in the proof of Corollary 5. | Post your referee report to D2L |

Friday, October 20 | (Fall break) | |

Monday, October 23 | Chapter 4.3: Recognize that every group acts on itself by conjugation. Understand the definition of conjugacy classes as orbits under this action. Be comfortable with the derivation of the so-called "class equation" (but don't worry about memorizing the equation itself). Understand the number theoretic cleverness involved in the proofs of the class equation and theorems 8 and 9. | Post your referee report to D2L |

Wednesday, October 25 | Chapter 4.3: Be very
comfortable with conjugacy in S_{n}, and know how to
compute the size of conjugacy classes. Understand the broad
strokes of how this is used in proving the simplicity of A_{n}
| Write up homework 8 |

Friday, October 27 | Chapter 4.4: Know what automorphisms and automorphism groups are, and understand that Aut(G) acts on G. Be very comfortable with the examples. | Post your problem to D2L |

Monday, October 30 | Chapter 4.5: Know what Sylow subgroups are, and be comfortable with the theory through the examples on page 142. | Post your referee report to D2L |

Wednesday, November 1 | Chapter 4.5: Understand how the Sylow theorems are useful in getting information about groups of small or medium order. Be confident that you could do the same, as in exercises 11 through 31. | Write up homework 9 |

Friday, November 3 | Chapter 5.1: Know what direct products are, and internalize the definitions and theorems here. Be aware that if H and K are normal subgroups of G with trivial intersection, such that every element of H commutes with every element of K, then HK is isomorphic to the direct product of H and K. | Post your problem to D2L |

Monday, November 6 | Chapter 5.2: Internalize the fundamental theorem, in both the prime factor and invariant factor version, and be able to pass between the two versions. Worry about being able to apply the theorem rather than proving it. It'll be a corollary of something else we won't prove in chapter 12. | Post your referee report to D2L |

Wednesday, November 8 | Chapter 5.3: Know most of the content of this table, and have at least a vague idea of the rest. | Don't write up homework 10 |

Friday, November 10 | Chapter 5.4: Internalize the statements of proposition 8 and theorem 9, about recognizing when an internal product HK is direct. | Post your problem to D2L |

Monday, November 13 | Chapter 5.5: Be very comfortable with the discussion on the first page or so, before Theorem 10. Recognize Theorem 10 as an awkward way to codify that discussion. | Post your referee report to D2L |

Wednesday, November 15 | Chapter 5.5: Understand the external construction of semidirect products, and its relationship to the internal semidirect product. Do not lose track of the internal meaning, and be able to write down a presentation for the external direct product of H and K, given presentations for H and K and a homomorphism from K to the automorphism group of H. | Write up homework 11 |

Friday, November 17 | Chapter 6.1: Know the major theorems involving p-groups, commutators, and solvability. | Post your problem to D2L |

Monday, November 20 | Chapter 7.1: Know the definition
of a ring. Be comfortable with the definitions of the major
variants (commutative ring, ring with 1, integral domain, field),
and know examples of rings which are (and aren't) of each type.
Make sure you have internalized all the statements of Proposition 1, and are able to prove them. Then be comfortable with all the examples. | Post your referee report to D2L |

Wednesday, November 22 | (Thanksgiving) | |

Friday, November 24 | (Thanksgiving) | |

Monday, November 27 | Chapter 7.2: Be very comfortable with the examples. | Post your referee report to D2L |

Wednesday, November 29 | Chapter 7.3: Understand that the definitions and constructions involved in homomomorphisms of rings and quotient rings are analogous to those for groups. Recognize that ideals are subrings which occur as kernels of homomorphisms, and so serve as the analog of normal subgroups. Observe that the isomorphism theorems for rings are verbatim copies of the isomorphism theorems for groups. | Write up homework 12 |

Friday, December 1 | Chapter 7.4: Know the basic definitions, properties, and examples of ideals very well. Internalize the statments relating prime and maximal ideals to domains and fields. | Post your problem to D2L |

Monday, December 4 | Chapter 7.5: Be very comfortable
with the construction of a ring (or field) of fractions from a
domain, and understand that this is the algebraic construction
of Q from R. Make sure you won't get confused
between this construction (a ring of fractions or quotients) and
the quotient ring construction (involving cosets).
| Post your referee report to D2L |

Wednesday, December 6 | Chapter 7.6: Know the (ring-theoretic) statement of the Chinese Remainder theorem, including the necessary definitions. | Write up homework 13 |

Friday, December 8 | Chapter 7.6: Get ahold of a number theory text, and understand how the (number-theoretic) Chinese Remainder Theorem is used to solve simultaneous congruences. Be clear how this is a special case of the (ring-theoretic) CRT. |