Math 0430 - Intro to Abstract Algebraic Systems (Fall 2017)

Syllabus
Schedule

Contact Info

roed@pitt.edu

prm50@pitt.edu

Textbook

Tom Judson, Abstract Algebra: Theory and Applications. It is available online [Printable] for free, or for sale ($26 hardcover or $3 kindle). There is also an extended version available that includes Sage exercises.

Lecture

Days	Time	Location	CRN
Tue Thu	16:00-17:15	Posvar 5401	11819

Office Hours

My office is Thackeray 426. I will hold office hours Tuesday and Thursday 2:00-3:45 and am also available by appointment.

Exams

The midterm will be on October 24 and the final will be on December 15. More study materials will be posted as the semester progresses.

Practice Midterm	Practice Midterm Solutions
Practice Final	Practice Final Solutions
2016 Midterm	2016 Midterm Solutions

Problem Sets

Due	Problems	Solutions	LaTeX Solutions
Sep 7	Problem Set 1	PDF Solutions	LaTeX Solutions
Sep 21	Problem Set 2	PDF Solutions	LaTeX Solutions
Oct 5	Problem Set 3	PDF Solutions	LaTeX Solutions
Oct 19	Problem Set 4	PDF Solutions	LaTeX Solutions
Nov 2	Problem Set 5	PDF Solutions	LaTeX Solutions
Nov 16	Problem Set 6
Dec 5

Daily Homework

Due	Problem
Aug 31	Find a relation on $\mathbb{Z}$ (the integers) that is transitive, NOT symmetric and NOT reflexive.
Sep 5	Find all of the partitions of the set $\{1,2,3,4\}$.
Sep 7	A primitive element modulo $n$ is a number $a$ so that the powers of $a$ modulo $n$ cover all of the conjugacy classes that are relatively prime to $n$ (for example, 2 is a primitive element modulo 5). Find an $n$ where a primitive element does not exist.
Sep 12	Find an example of a two elements $a, b$ in a group and an integer $n$ so that $(ab)^n \ne a^nb^n$.
Sep 14	Find all of the subgroups of $D_4$.
Sep 19	Find the smallest subgroup of $\mathbb{Z}/{32}$ containing both $12$ and $18$
Sep 21	Find a primitive $12$th root of unity (in the form $a + bi$).
Sep 26	List the different cycle types in $S_7$ and give the order for an element of each type.
Sep 28	Find all of the subgroups of $A_4$.
Oct 3	Find the right and left cosets within $A_4$ of the subgroup generated by $(123)$.
Oct 5	No daily homework
Oct 12	Find an isomorphism between $\mathbb{Z}_6$ and $U(18) = \mathbb{Z}_{18}^\times$.
Oct 17	No daily homework
Oct 19	For each subgroup $H$ of $Q_8$, determine whether or not it's normal, and if it is, identify the quotient $Q_8/H$ from the list $\{1\}, \mathbb{Z}_2, \mathbb{Z}_4, \mathbb{Z}_2 \times \mathbb{Z}_2, Q_8$.
Oct 24	No daily homework
Oct 26	No daily homework
Oct 31	Find a nontrivial homomorphism from $U(8)$ to $U(18)$.
Nov 2	No daily homework
Nov 7	Find an example of two integral domains $R$ and $S$ together with a zero divisor in $R \times S$, showing that the direct product of integral domains is not necessarily an integral domain.

Extra Credit Projects

There are three options for extra credit. They can contribute up to 5% to your final grade.

Group theory project.
Sage project.
Rings project. If you want to learn about class groups and algebraic number theory, contact me and I will provide a project and guidance.

LaTeX

Here are some resources for learning and improving your LaTeX skills.

George Grätzer, A Short Course in LaTeX.
Wikibooks, LaTeX Guide.
Tobias Oetiker, The Not So Short Introduction to LaTeX2ε.
Vincent Knight, videos on using LaTeX in Sage Math Cloud.

Sage

Sage is an open source mathematics software package that is capable of computing with many of the objects that we will be learning about this semester. Moreover, there is a version of our textbook that includes sections on using Sage and Sage exercises.

If you are interested in learning about Sage, you can do the Sage extra credit project. Here are some resources (beyond what is in the textbook) for learning about Sage.

You can create an account on CoCalc and use Sage online, or download it here.
The Online version of the textbook is interactive, with examples that you can play around with.
The Sage tutorial is another way to get started with Sage.
This video, by the founder of Sage (and Sage Math Cloud), has a bunch of useful tips, though it's a bit long.

General Study Skills

The Academic Resource Center holds various study skills workshops, as well as individual consulations. See here for more details.

Due	Problem
Aug 31	Find a relation on \(\mathbb{Z}\) (the integers) that is transitive, NOT symmetric and NOT reflexive.
Sep 5	Find all of the partitions of the set \(\{1,2,3,4\}\).
Sep 7	A primitive element modulo \(n\) is a number \(a\) so that the powers of \(a\) modulo \(n\) cover all of the conjugacy classes that are relatively prime to \(n\) (for example, 2 is a primitive element modulo 5). Find an \(n\) where a primitive element does not exist.
Sep 12	Find an example of a two elements \(a, b\) in a group and an integer \(n\) so that \((ab)^n \ne a^nb^n\).
Sep 14	Find all of the subgroups of \(D_4\).
Sep 19	Find the smallest subgroup of \(\mathbb{Z}/{32}\) containing both \(12\) and \(18\)
Sep 21	Find a primitive \(12\)th root of unity (in the form \(a + bi\)).
Sep 26	List the different cycle types in \(S_7\) and give the order for an element of each type.
Sep 28	Find all of the subgroups of \(A_4\).
Oct 3	Find the right and left cosets within \(A_4\) of the subgroup generated by \((123)\).
Oct 5	No daily homework
Oct 12	Find an isomorphism between \(\mathbb{Z}_6\) and \(U(18) = \mathbb{Z}_{18}^\times\).
Oct 17	No daily homework
Oct 19	For each subgroup \(H\) of \(Q_8\), determine whether or not it's normal, and if it is, identify the quotient \(Q_8/H\) from the list \(\{1\}, \mathbb{Z}_2, \mathbb{Z}_4, \mathbb{Z}_2 \times \mathbb{Z}_2, Q_8\).
Oct 24	No daily homework
Oct 26	No daily homework
Oct 31	Find a nontrivial homomorphism from \(U(8)\) to \(U(18)\).
Nov 2	No daily homework
Nov 7	Find an example of two integral domains \(R\) and \(S\) together with a zero divisor in \(R \times S\), showing that the direct product of integral domains is not necessarily an integral domain.