ETSU Abstract Algebra Club
Our "logo" which gives the relationship between a normal subgroup H and its cosets in group G
and the canonical homomorphism φ from G onto φ[G].
The ETSU Abstract Algebra Club is an unofficial, unrecognized organization created by Robert "Dr. Bob" Gardner in 2014 to promote abstract (or modern) algebra and the classes which ETSU offers in this area. ETSU offers two undergraduate/graduate level cross-listed classes, Introduction to Modern Algebra (MATH 4127/5127) and Introduction to Modern Algebra 2 (MATH 4137/5137), and two graduate-only level classes, Modern Algebra 1 (MATH 5410) and Modern Algebra 2 (MATH 5420).
Membership is open to all ETSU students interested in abstract algebra. Perks of membership include the right to attend club meetings and presentations, and an unofficial membership card:
If you would like to join, then send an e-mail to Dr. Bob (email@example.com) and he will (eventually) get around to having a membership card printed up for you.
Textbooks, Class Notes, and Syllabi
The textbooks currently (academic 2014-15) used in the modern algebra classes at ETSU are:
You can find notes based on these books online:
Some syllabi are online for some of our modern algebra classes:
A First Course in Abstract Algebra, 7th edition,
by John B. Fraleigh (2002)
by Thomas W. Hungerford (1974)
- Introduction to Modern Algebra (MATH 4127): Spring 2013, Fall 2013, and Fall 2014.
- Introduction to Modern Algebra (MATH 5127): Spring 2013, Fall 2013, and Fall 2014.
- Introduction to Modern Algebra 2 (MATH 4237): Spring 2014, and Spring 2015.
- Introduction to Modern Algebra 2 (MATH 5237): Spring 2014, and Spring 2015.
- Modern Algebra 1 (MATH 5410): Fall 2015, Spring 2017, Fall 2017, Fall 2019, Spring 2021, and Fall 2021.
- Modern Algebra 2 (MATH 5420): Spring 2016, and Spring 2018.
The following online presentations may be of interest to club members:
Some supplemental class notes which might also interest members (but is a bit more technical than the above presentations) include:
"The Bicentennial of Evariste Galois," a historical discussion of one of the founders of abstract algebra by Dr. Bob: Website to link to presentation and photos. A YouTube video of the presentation is here. (accessed 1/2/2015)
"Permutation Groups: Cycles, Transpositions, and Futurama," an introduction to permutation groups and an application in the TV show Futurama by Dr. Bob: PowerPoint.
- "Groups: A Geometric Introduction," is an introduction to frieze and wallpaper groups by Dr. Bob. There are some undefined terms in the presentation (such as "subgroup," "isomorphism", and "coset") which are briefly explained. This is also a supplement to Section II.12 of Fraliegh's text. There is a PowerPoint presentation (prepared in Microsoft PowerPoint 2010) with embedded audio, a transcript in PDF, and a Windows Media Video: PowerPoint, transcript, video. A corrected version of the PowerPoint is also available (but it lacks the embedded audio). The video is also on YouTube here (accessed 9/22/2017).
"The Quaternions: An Algebraic Approach," a talk on the noncommutative division ring the quaternions. This was part of the club's 175th anniversary celebration of Sir William Rowan Hamilton's introduction of the quaternions in 1843. The talk is available in PowerPoint.
"Integral Domains and Fields of Quotients," a talk that extends an integral domain to a field by defining addition and multiplication on equivalence classes of ordered pairs of elements of the integral domain. This talk covered the content of the club t-shirts for fall 2021.
The talk is available in PowerPoint and a
Zoom recording is also available.
"Numbers: The Reals and Beyond!", a talk that starts with the field of real numbers, and constructs the complex numbers from the reals. The complex field is the is the algebraic closure of the reals. We will proceed beyond the complex numbers with the Cayley-Dickson construction and consider the quaternions, octonions, and sedenions. This talk covered the content of the club t-shirts for fall 2023.
The talk is available in PowerPoint.
- "Sylow Sesquicententennial," a talk in commemoration of the 150th anniversary of the publication of Peter Ludwig Sylow's 1872 paper in which he presented his famous three theorems. The anniversary was also marked with the fall 2022 club t-shirts.
The talk is available in PowerPoint.
- "Group Theory and the Rubik's Cube," an application of group theory to the permutations of Rubik's cube, by Dr. R.B. Beeler: PDF.
- "Graph Automorphism Groups," a discussion of Cayley graphs and automorphism groups of graphs, by Dr. R.B. Beeler (updated fall 2022): PDF.
- "Polya: But I Hardly Know Ya!," applications of Burnside's Lemma and Polya's Enumeration Theorem to various graph colorings, by Dr. R.B. Beeler: PDF.
From Introduction to Modern Algebra (MATH 4127/5127): A Student's Question: Why the Hell am I in this Class?. This is a motivational presentation which explains how "classical" algebra (polynomial equations) relates to "modern" algebra (group theory, ring theory, and field theory).
From Introduction to Modern Algebra (MATH 4127/5127): Small Groups.
From Introduction to Modern Algebra (MATH 4127/5127): Alternating Groups are Simple.
From Introduction to Modern Algebra (but useful to any abstract algebra fan): Finite Simple Groups.
From Introduction to Modern Algebra 2 (MATH 4137/5137): Algebraic Closure of a Field.
From Introduction to Modern Algebra 2 (MATH 4137/5137): Compass and Straight Edge Constructions. There is a PowerPoint presentation (prepared in Microsoft PowerPoint 2010) with embedded audio, a transcript in PDF, and a Windows Media Video: PowerPoint, transcript, video. The video is also on YouTube here (last accessed 7/16/2014).
From Introduction to Modern Algebra 2 (MATH 4137/5137): Section 7.4. Integral Quaternions and the Four-Square Theorem (based on Herstein's Topics in Algebra, 2nd Edition).
From Introduction to Modern Algebra 2 (MATH 4137/5137): Supplement The Cayley-Dickson Construction and Nonassociative Algebras (based on Herstein's Topics in Algebra, 2nd Edition).
From Modern Algebra 1 (MATH 5410): Direct Products and Semidirect Products (based on Sections 5.4 and 5.5 of Dummitt and Foote's Abstract Algebra, 3rd Edition; this requires some knowledge of automorphism groups).
From Modern Algebra 1 (MATH 5410): The Fundamental Theorem of Algebra-History. A more visual presentation (including an intuitive argument for the Fundamental Theorem) is also available in PowerPoint.
From Modern Algebra 1 (MATH 5410): The Fundamental Theorem of Algebra (an appendix to Section V.3 of Hungerford's Algebra text; this includes a mostly-algebraic proof of the Fundamental Theorem of Algebra, but requires a significant amount of background material in field theory).
From Modern Algebra 2 (MATH 5420): Information on the "Snake Lemma," made somewhat famous by being the topic of a lecture by a character portrayed by Jill Clayburgh in the movie It's My Turn costarring Michael Douglas (Rastar Films, 1980), is available in
Section IV.1. Modules, Homomorphisms, and Exact Sequences. A proof is given in a supplement to these notes in Supplement. A Proof of The Snake Lemma.
The club started holding official meetings during fall 2014. Activities have included presentations relevant to abstract algebra, pizza feasts, and the design of a t-shirt. For more details on club activities, click on the links below.
- Fall 2014 activities.
- Spring 2015 activities.
- Fall 2015 activities.
- Spring 2016 activities.
- Fall 2016 activities.
- Spring 2017 activities.
- Fall 2017 activities.
- Spring 2018 activities.
- Fall 2018 activities.
- Spring 2019 activities.
- Fall 2019 activities.
- Spring 2020 activities.
- Fall 2021 activities.
- Spring 2022 activities.
- Fall 2022 activities.
- Spring 2023 activities.
- Fall 2023 activities.
Bob Gardner's home page
Last updated: September 9, 2023.