Group Theory

Syllabus

Calendar

There is a proposed calendar on the syllabus, but here I will record what we actually get through in class.

  • …
  • 11/30 & 12/7 – We covered theorems that tell us about the structure of finite groups beyond just Abelian groups. In particular on the last night we looked at proofs of Cauchy’s theorem, the Class Equation, and Sylow’s First Theorem. In this unit we looked at new material from chapters 9, 13, 14, 17, and 19, as well as using the First Isomorphism theorem for which you should look at Theorem 12.14.
  • 11/16 – We looked at a small version of the Fundamental Theorem of Finitely Generated Abelian Groups so that the full version in the text would be easier to understand. We then looked at ways we can apply this theorem to be able to list all possible abelian groups of a given order: e.g. if \(G\) is abelian and has order \(p^2q\) where \(p\) and \(q\) are prime, then we can say that either \(G\cong \mathbb{Z}_{12}\) or \(G\cong \mathbb{Z}_2\times\mathbb{Z}_6\), which are not isomorphic since the first has an element of order 4 and the second does not.
  • 11/9 – Looking at Fraleigh’s treatment of the Fundamental Theorem of Finitely Generated Abelian Groups, it makes more sense to build toward that proof before looking at Cauchy’s Theorem or Sylow’s Theorems. So tonight we started building toward that and will cover the main theorem next week.
  • 11/2 – We discussed ways in which maps can induce a structure on the domain from the codomain, Normal Subgroups, Homomorphisms, and The First Isomorphism Theorem.
  • 10/26 – Exam on Basic Group Properties
  • 10/19 – In a whirlwind lecture through these slides Groups and Homomorphisms and Cosets and Lagrange, we looked at homomorphisms, permutations, Cayley’s theorem, cosets, and Lagrange’s Theorem. This unit we have covered material from sections 1-6, 8, 9, and 10. We didn’t get to 11 and 12, 12 will be pushed to the next unit and we’ll drop 11. For the exam next week look at the study guide below. For section 10 the only questions on the exam will be computation questions which focus on finding left and right cosets as we discussed in class tonight and finding indexes of subgroups.
  • 10/12 – We spent most of the night looking at material from these slides: Groups and Subgroups. We then spent a little time looking at these Groups and Homomorphisms, we will continue with these next week.
  • 10/5 – We looked at some specific examples of groups, the dihedral group \(D_n\) and the symmetric group \(S_n\). I also mentioned that all abelian groups look like combinations of \(\mathbb{Z}\) or \(\mathbb{Z}_n\), what I should have said is all finitely generated abelian groups look that way; \(\mathbb{Q}\) and \(\mathbb{R}\) are abelian but do not look that way.
  • 9/28: Exam 1
  • 9/21: Rings, Ideals, and Quotients
  • 9/14 – Tonight, after the quiz, we reviewed some linear algebra. In particular we discussed Linear Transformations\[T:V\rightarrow W\ with\ T(a\vec{v}_1+b\vec{v}_2)=aT(\vec{v}_1)+bT(\vec{v}_2),\] Null Spaces\[N=\{\vec{v}\in V | T(\vec{v})=\vec{0}_W,\}\subseteq V\] and Column Spaces \[T(V)=\{T(\vec{v})|\vec{v}\in T\}\subseteq W.\] We also pointed out that the dimension of the null space plus the dimension of the column space is always equal to the number of columns of the matrix of the transformation and, finally, we can define an equivalence relation on the domain \(T\) using the null space which respects the arithmetic in \(T\). We looked at all of this so that when we went back to rings we could recognize that we are studying analogous structures that behave in the same way. Ring homomorphisms \[\phi:R\rightarrow S\ with\ \phi(a+b)=\phi(a)+\phi(b)\ and\ \phi(ab)=\phi(a)\phi(b),\] Kernels\[K=\{r\in R | \phi(r)=0_S,\}\subseteq R\] and Images \[\phi(R)=\{\phi(r)|r\in R\}\subseteq S.\] We ended by pointing out that the kernel is a special type of subring called an ideal. We will pick up the discussion there next week.
  • 9/7 – Tonight we had a quiz on equivalence relations and discussed relations a little more in terms of the quiz questions and the packet from last week. Then we looked at sections 1,2,4,5,7, and 8 in the Algebraic Structures Overview packet. You should review this and play around with sections 3 and 6 ahead of the quiz next week.
  • 8/31 – Today we went over the syllabus and then worked through section 2 and part of section 3 of the Exploring Equivalence Relations packet. Please finish section 3 to make sure that you don’t have any questions. You may complete section 4 for extra assignment credit; it is due by the end of September.

Unit 1: Algebraic Structures

Assignment Due 9/28/2023:

All assignment submissions must be typed and in complete sentences. Proper submission formatting may count for up to 10% of the assignment grade. You need to do four problems total (3 concept and 1 theory), no two problems may be from the same section.

Concept Questions:

Answer three of the following

  • Sec. 2 :24,26
  • Sec. 22: 34
  • Sec. 23: 22, 24
  • Sec. 30: 8, 12

Theory Questions:

Answer one of the following

  • Sec. 2: 38, 42
  • Sec. 22: 48, 50
  • Sec. 23: 32, 36
  • Sec. 30: 18, 32

Test Preparation Materials for Exam on 9/21/2023:

Definitions:

Equivalence Relation and Class, Modular Equivalence, Relatively Prime, Functions, One-to-One, Onto, Group, Ring, Integral Domain, Field, Subring, Ideal, Characteristic, Zero Divisor, Unit, Homomorphisms, Kernels. Quotient Structures

Theorems:

  • Theorem 2.18: Identity and Inverse Elements are unique.
  • Corollary 23.5-ish: \(\mathbb{Z}_p\) is a Field if and only if \(p\) is prime.
  • Theorem 23.11: Every finite integral domain is a field.
  • Theorem 30.11: Properties of Homomorphisms.
  • Theorem 30.14: A homomorphism \(\phi\) is 1-1 if and only if \(ker_\phi=\{0\}\).
  • Theorem 30.15: For a homomorphism \(\phi\), \(ker_\phi\) is an ideal.

Practice Problems:

For each exam look at the odd numbered Computation questions, the correct the definition Concept questions, the True/False Concept questions, and, when they are there, the Proof Synopsis questions.

Unit 2: Groups, Properties, Homomorphisms

Assignment Due 11/2/2023:

All assignment submissions must be typed and in complete sentences. Proper submission formatting may count for up to 10% of the assignment grade. You need to do four problems total (3 concept and 1 theory), no two problems may be from the same section.

Concept Questions:

Answer three of the following

  • Sec. 2: 24
  • Sec. 4: 28
  • Sec. 5: 40
  • Sec. 6: 46
  • Sec. 7: 20
  • Sec. 8: 30, 32
  • Sec. 10: 22, 26
  • Sec. 12: 22

Theory Questions:

Answer one of the following

  • Sec. 4: 36
  • Sec. 5: 56
  • Sec. 6: 56, 66
  • Sec. 7: 22
  • Sec. 8: 38
  • Sec. 10: 36
  • Sec. 12: 32, 36

Test Preparation Materials for Exam on 10/26/2023:

Definitions:

Abelian Group, Non-Abelian Group, Dihedral Group, Permutation Group (or Symmetric Group), Cyclic Group, Subgroup, Cycles, Permutations, Even and Odd Permutations, Homomorphism, Isomorphism, Aut(G), Inn(G), Coset, Normal Subgroup, Factor Group (or Quotient Group), Order of a Group, Order of an Element, Center of a Group, Centralizer of an Element, Generator(s), Normalizer, Index of a Subgroup

Theorems:

  • Theorems 5.12 or 5.15 (variations on the two-step test)
  • Properties of Orders – Look at the slides and just focus on the ones presented there.
  • Theorem 8.5 Properties of Homomorphisms
  • Definition 8.10, Theorem 8.11: Cayley’s Theorem
  • Properties of Cosets – Look at the slides and Theorem 10.6 in Particular
  • Theorem 10.7: Lagrange’s Theorem
  • First Isomorphism Theorem

Practice Problems:

For each exam look at the odd numbered Computation questions, the correct the definition Concept questions, the True/False Concept questions, and, when they are there, the Proof Synopsis questions. For section 10 the only questions on the exam will be computation questions which focus on finding left and right cosets as we discussed in class tonight and finding indexes of subgroups.

Unit 3: Advanced Topics in Group Theory

Assignment Due 12/15/2023:

All assignment submissions must be typed and in complete sentences. Proper submission formatting may count for up to 10% of the assignment grade. You need to do four problems total (3 concept and 1 theory), no two problems may be from the same section.

Concept Questions:

Answer three of the following

  • Sec. 9: 34, 38 (see example 9.13)
  • Sec. 12: 22
  • Sec. 13: 22, 26, 28
  • Sec. 14: 14, 18

Theory Questions:

Answer one of the following

  • Sec. 12: 32, 36
  • Sec. 16: 8, 10 – Bonus credit for doing one of these.
  • Sec. 17: 24
  • Sec. 19: 16

Test Preparation Materials for Exam on 12/14/2023:

Definitions:

Isomorphism, Aut(G), Inn(G), Homomorphism, Kernel, Conjugacy Class, p-Group, Sylow p-Subgroup, Generators, Relation, Equivalence Classes

Theorems:

  • Isomorphism Theorems
  • First Isomorphism Theorem (Theorem 12.14)
  • Lagrange’s Theorem (Theorem 10.7)
  • Fundamental Theorem of Finitely Generated Abelian Groups (Theorem 19.12, but also look at 19.8, 19.9, and 19.11 to get a good understanding of the proof)
  • Classification of Groups of order \(p^2\) (Theorem 14.25)
  • Cauchy’s Theorem (Handout or Theorem 14.20)
  • Sylow’s First Theorem (Theorem 17.4)

Practice Problems:

For each exam look at the odd numbered Computation questions, the correct the definition Concept questions, the True/False Concept questions, and, when they are there, the Proof Synopsis questions.

Typesetting Out-of-Class Work (Template)

  1. Create an account at Overleaf.com using your WCSU email account
  2. Watch the introductory video here VIDEO LINK, the document created in the video is here: https://www.overleaf.com/read/khbktbvvvkjd
  3. Follow this link to see samples of typed up exercises: https://www.overleaf.com/read/frptnykkgpbt
  4. Follow this link to make a copy of a blank template: https://www.overleaf.com/read/hwdhcvmntrvs
  5. Submit work by downloading a copy of the PDF you generate and printing it.

If you are having problems typing up a document and would like feedback you can share a link to it by following the directions here: https://www.overleaf.com/learn/how-to/Sharing_a_project; send me the link to view but not edit. Here is a quick reference sheet with symbols and such: Quick Reference. Finally, here is a playlist of video lessons on \(\LaTeX\): Technical Typesetting with \(LaTeX\) Playlist

Links and Handouts