Math 448: Modern Algebra I

Assignment #24 – Tuesday, Nov 30, 2021

More Fun with Groups!

Write up and hand in the following.

  1. Let $ABCDEF$ be a regular hexgon centered at the origin with point $A$ located at $(1,0)$ and point $B$ in the first quadrant. Let $r$ be the symmetry operation of $ABCDEF$ that rotates it by an angle of $60^\circ$ about the origin in the counter-clockwise direction (i.e., it sends $A$ to $B$, $B$ to $C$, and so on). Let $m$ be the reflection of $ABCDEF$ about the $x$-axis (i.e. it sends $A$ to $A$, $B$ to $F$ and so on). If $a$ and $b$ are symmetry operations, $ab$ is composition of those operations as functions, so that $ab$ means “first do $b$ then do $a$ in that order”. In this notation the dihedral group, $D_6$ consists of the elements $$\{e,r,r^2,r^3,r^4,r^5,m,mr,mr^2,mr^3,mr^4,mr^5\}$$
    1. Write out the multiplication table for $D_6$ in terms of these 12 elements (i.e. only those 12 elements can appear in your table).
    2. Determine the order of each of the 12 elements of $D_6$. Note: this is level “Easy” in Scrambler! (at the ToyProofs link below).
  2. Exercise 7.2 #12.
  3. Exercise 7.2 #18.
  4. Exercise 7.2 #24. Hint: One transitive chain, man!
  5. Exercise 7.2 #36. Warning: This problem is fun!

Course Handouts and References

  • Lecture Notes – check here often for revised lecture notes for our course
  • Course Syllabus
  • Proofs – notes and handouts from my Math 299 course on mathematical proofs

 Math 448 Online

  • Math 448 Online Classroom
  • If we have to pivot to remote instruction this semester, click here to enter our Zoom classroom.
  • Zoom help – Zoom for Scranton students, how to install, how to use, documentation.

Software and Handouts

  • Lurch – a math word processor that can check your proofs! I designed and wrote this software with former Scranton math major Nathan Carter
  • Overleaf – a free website where you can easily produce LaTeX math documents through a web browser
  • Toy Proofs – a “toy” proof system I developed to introduce students to the concept of formal proofs.
  • Toy Proof Talk Slides – slides from a talk I gave on Toy Proofs at the 2009 Joint Mathematics Meetings
  • Group Explorer – amazing group visualization and exploration software from Nathan Carter

Homework Assignment Template

Tentative Schedule
1 Tue, Aug 31 Introduction
2 Thu, Sep 2 Logic and Proof
3 Tue, Sep 7 Appendix B: Sets and Functions
4 Thu, Sep 9 Appendix D: Equivalence Relations
5 Tue, Sep 14 Appendix C: Induction
6 Thu, Sep 16 Section 1.1: Division Algorithm
7 Tue, Sep 21 Section 1.2: Divisibility
8 Thu, Sep 23 Section 1.3: Primes and Unique Factorization
9-10 Tue, Sep 28 Section 2.1-2.2: Congruence and Modular Arithmetic
11 Thu, Sep 30 Section 2.3: $\mathbb{Z}_p$ when $p$ is prime
12 Tue, Oct 5 Section 3.1: Rings
13 Thu, Oct 7 Section 3.2: Basic Properties of Rings
14 Thu, Oct 14 Section 3.3: Isomorphism and Homomorphism
15 Tue, Oct 19 Section 4.1: Polynomials and Division Algorithm
16 Thu, Oct 21 Section 4.2: Divisibility in $F[x]$
17 Tue, Oct 26 Section 4.3: Irreducibles and Unique Factorization
18 Thu, Oct 28 Section 4.4: Polynomial Functions, Roots, and Divisibility
19 Tue, Nov 2 Section 5.1: Congruence in $F[x]$
20 Thu, Nov 4 Section 5.2: Modular Arithmetic in $F[x]$
21 Tue, Nov 9 Section 5.3: $F[x]\left/p(x)\right.$ when $p(x)$ is Irreducible
22 Thu, Nov 11 Section 6.1: Ideals and Congruence
23 Tue, Nov 16 Section 6.2: Quotient Rings and Homomorphisms
24 Thu, Nov 18 Section 7.1: Groups
25 Tue, Nov 23 Section 7.2: Basic Properties of Groups
26 Tue, Nov 30 Section 7.3: Subgroups
27 Thu, Dec 2 Section 7.4: Isomorphisms and Homomorphisms & Putnam!
28 Tue, Dec 7 Section 8.1: Congruence and Lagrange’s Theorem
29 Thu, Dec 9 Section 7.5: Symmetric and Alternating Groups