Homework assignments for Jeff Mermin's Math 5003 class

Wednesday, August 23Skim 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 25Chapter 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 28Chapter 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 30Chapter 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 1Chapter 1.4: Understand the definition of the general linear group, GLn(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 4Labor day: cast aside the yoke of oppression. Post your referee report to D2L. (This deadline is fluid, because Labor day.)
Wednesday, September 6Chapter 1.5: Know the quaternion group. Then reread chapter 1.3. Write up homework 2
Friday, September 8Chapter 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 11Chapter 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 13Chapter 1.7: Be very comfortable with the prose discussion of group actions, and with all five examples. Write up homework 3
Friday, September 15Chapter 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 18Chapter 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 20Chapter 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 22Chapter 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 25Chapter 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 27Chapter 2.5: Make sure you're comfortable reading these lattices.
Friday, September 29Chapter 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 2Chapter 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 4Chapter 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 6Chapter 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 9Chapter 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 11Chapter 3.5: Know what transpositions are, and know that Sn is generated by its transpositions. Understand that a homomorphism, called "sign", from Sn 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 13Chapter 3.5: Know at least one definition of the sign homomorphism, and the definition of the alternating group An. Understand that An is normal in Sn, and know its role in composition series for Sn. Post your problem to D2L
Monday, October 16Chapter 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 SA. Post your problem to D2L
Wednesday, October 18Chapter 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 23Chapter 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 25Chapter 4.3: Be very comfortable with conjugacy in Sn, and know how to compute the size of conjugacy classes. Understand the broad strokes of how this is used in proving the simplicity of An Write up homework 8
Friday, October 27Chapter 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 30Chapter 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 1Chapter 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 3Chapter 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 6Chapter 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 8Chapter 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 10Chapter 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 13Chapter 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 15Chapter 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 17Chapter 6.1: Know the major theorems involving p-groups, commutators, and solvability. Post your problem to D2L
Monday, November 20Chapter 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 27Chapter 7.2: Be very comfortable with the examples. Post your referee report to D2L
Wednesday, November 29Chapter 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 1Chapter 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 4Chapter 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 6Chapter 7.6: Know the (ring-theoretic) statement of the Chinese Remainder theorem, including the necessary definitions. Write up homework 13
Friday, December 8Chapter 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.