# What are some good mathematical applications to present in an abstract algebra course?

One of the main difficulties for a student learning abstract algebra is understanding the motivations behind concepts like groups, normal subgroups, rings , ideals etc. Also, many have difficulty understanding why algebra is important and how it has helped pure mathematics or is applied in mathematics.

So, I would like to know what are some of the nice applications of algebra that can be taught in first algebra course.

Two examples that come to mind are unsolvabillity of equations of degree greater or equal to 5 by radicals and algebraic number theory. But it is difficult to include them in a first algebra course due to time constraints.

Are there any other applications of algebra which can be introduced easily in a first algebra course?

• Though this is a good question, the title is very generic. Perhaps it should be changed to something like "What are some good applications to present in an abstract algebra course?" – Jim Belk Mar 16 '14 at 17:28
• In Frenkel's autobiographical "Love and Math" there is an accessible (for students with a lot less background) discussion going from equations to groups to group actions to Lie groups to Loop groups and some nice pictures to communicate the landscape of math. It might give you some ideas. As a student, the application of the orbit stabilizer theorem to count symmetries of objects was deeply interesting for me. Maybe a good approach is to have a list of applications where you just explain it in say three sentences. An invitation to future reading. That's more tenable than something with teeth – James S. Cook Mar 16 '14 at 18:24
• Taking about the insolvability of the quintic in an introductory course is certainly possible. Just this past fall I taught an 10 week course for students for whom this was going to be the only abstract course they would ever take, so I had complete control over the content. We spent 5 weeks on groups, 2 on polynomial rings, 2 on field extensions, and 1 on Galois theory. By the end we were able to give (with complete proof) an explicit example of a insolvable quintic, and along the way they learned a good amount of group theory. – Santiago Canez Mar 16 '14 at 20:57
• I want to point out that showing specific examples of groups or rings might not constitute an "application of abstract algebra". The most difficult thing for students to swallow when they first meet abstract algebra, I think, isn't that specific examples of groups or rings are interesting, it's that there's a good reason to create a general theory of groups and rings. Even if groups show up in two, three, or five different places, is that a sufficient reason to create a totally general theory of groups? That's the point that needs to be explained, I think. – Jack M Apr 12 '14 at 21:48
• Group theory is a big deal in inorganic chemistry (point groups, space groups, IR bending modes, etc.) See for example: amazon.com/Chemical-Applications-Group-Theory-3rd/dp/0471510947 and many similar competing texts. Also the crystallographic tables (it's a book). I'm kind of in shock that an instructor would not know about this. Then again, maybe there is a reason why chemists and physicists never bother with courses in abstract algebra and instead look at books like Cotton's. – guest Oct 2 '18 at 9:33

I think a nice way to introduce groups is the Rubik's Cube non-commutative group. Also tessellation can be a way to inspire the need of a simple underlying structure to represent some complex sets. Also the dihedral group has the advantage of being visual.

As dtldarek pointed out, you can praise finite fields for their applications in asymmetric cryptography (RSA) or error-correcting codes like Reed-Solomon with their application to CDs. Or more generally the existence and ease of computing the inverse of any element of a finite field using the extended Euclidean algorithm.

My favorite hidden application of algebra is the Spot It! game, which uses projective planes over finite fields.

• Spot It! is similar to Set, which I think is more widespread. The Spot It! deck should have another card, if I recall correctly, which would make for a very nice discussion (which card, is there room for 2 more cards, etc). – Kevin O'Bryant Mar 25 '14 at 13:44
• Here's a link to a discussion about the (two) missing cards in Spot It!: plus.google.com/107909926350520444591/posts/amAe7JaHpdt , written by Michael Kleber. – Kevin O'Bryant Mar 25 '14 at 13:45

Burnside's lemma can be covered in one day (or less) in an abstract algebra course, and has lots of wonderful applications to counting problems.

• Yes indeed! A beautiful application is counting the number of necklaces. – Jill-Jênn Vie Mar 16 '14 at 23:36

You ask for applications of abstract algebra. One obvious application is in physics (Just google "group theory and physics).

But I also wanted to point out that one doesn't necessarily need applications to motivate a topic. If you want to motivate abstract algebra, you could start by discussing the operations you can perform with numbers. You can show that under multiplication, only $1$ and $-1$ have inverses in the natural numbers. But in the rational numbers all non-zero numbers have multiplicative inverses. This all leads very naturally to the definition of groups, rings, and fields. You could look at the integers $\mathbb{Z}$ modulo a given number $n$ (and then you have clock arithmetic). A very natural question then becomes: for which $n$ do we get a field?

My point is that motivation of many mathematical concepts can come from the fact that they arise so naturally. You show something that is already familiar and then you ask a slightly more abstract question and then boom you have all of abstract algebra (exaggerating a bit here).

• Well, not only I want to motivate the topics but I also want students to feel that algebra is a very tool without modern mathematics would have been impossible. – user774025 Mar 17 '14 at 3:14
• @user774025: And I think that when you explain that at some level mathematics is about abstraction and that by asking these slightly more abstract questions many things come very naturally, then that will get the point across. So all I wanted to do was to point this out because the other answers seem to address more specifically your request for applications. – Thomas Mar 17 '14 at 3:16

As dtldarek said telecommunications applications offer quite a few such examples. To make this more explicit I mention the following two (that I have tried in the past):

• Diffie-Hellman key exchange. After you have let the students prove that the group $\Bbb{Z}_p^*$ is cyclic (either always or for some specific suitable prime such as $p=107$ or $p=257$), then you can make them wonder, why the problem of solving $m$ from the congruence $$ma\equiv b\pmod{p-1}$$ for a given $a$, coprime to $p-1$ and $b$ is easy, but the "isomorphic" problem of solving $m$ from the congruence $$g^m\equiv b\pmod p$$ is difficult for a larger $p$ ($g$ a generator of $\Bbb{Z}_p^*$).
• Alamouti coding. This is a method of adding diversity to radio transmission. Transmission from a single antenna will occasionally fade (a moving receiver will occasionally be in a node when a signal combines destructively with a reflected copy of itself). This technique uses two separate antennas transmitting the signal intelligently such that the signals of the two antennas always superpose constructively. It is based on a clever but relatively easy to describe application of the algebra of the division ring of Hamiltonian quaternions.

My students commented on these in their teacher evaluation forms. The response was not unanimously positive (that will never happen), but some did say that such examples convinced them about the usefulness of abstract algebra.

I have mixed feelings about investing a lot of time on a course to examples like this. They do distract the students from the main material a little bit, they are not everybody's cup of tea, and the students may panic into thinking that they need to understand the example fully. The way I did it was to craft a few homework problems to serve as pons asinorum. Then in the homework problem sessions I took the stage (after a student had presented their solution), told them not to take notes, and spent may be 15-20 minutes outlining the application. I have enough control over my problem sessions so that this won't cause the session to run into overtime.

I also use permutation games as examples. 10-15 years ago a significant fraction of my students had played Rotation on their cellphones. Its group theory is very simple. I am retrying it this Spring. I don't expect it to be a hit as the game is probably quite passé now.

• If you don't want to distract from the course too much you could always talk about Diffie-Hellman using groups (you then get to talk about centralisers and suchlike). – user1729 Mar 27 '14 at 18:02

I'm not sure if this is what you want, but

• monoids are used in automata theory,
• semirings and matroids are used in graph theory,
• groups and finite fields are used in cryptography and coding theory.

The examples are like this, because abstract algebra is frequently tied to the discrete world.

Nevertheless, I hope this helps $\ddot\smile$

The book "Adventures in Group Theory" by David Joyner develops abstract algebra and uses it to "study certain toys and games from a mathematical modelling perspective". The main example is the Rubik's Cube, but there are plenty of others. For example, Chapter 4 is entitled "A procession of permutation puzzles" and looks at:

• Sam Loyd's 14-15 puzzle. This is a 4x4 sliding grid with tiles numbered 1 to 15, and the bottom right square is empty. The 14 and the 15 have switched places, and the "challenge" is to solve it.

• The hockeypuck puzzle. Cut a hockeypuck into 12 pieces as you would a cake (but using a saw as opposed to a knife...), and number them. You can flip the pie along a cut, and you can rotate it (by multiples of $\pi/6$). Solve it.

• Rainbow Masterball

• Pyraminx

• The $2^3$ and $3^3$ Rubik's cubes.

• Skewb

• Megaminx

I am unsure how it is as a set text or as an introduction to group theory. However, as a source of ideas and examples it is very good! Also, the text is full of SAGE examples, which you may or may not consider to be a good thing. However, if you want to teach someone SAGE or similar then using these kinds of puzzles and games is perhaps a good way.

Finally, using puzzles allows you to (tentatively) introduce group actions. Which is a very important way of looking at groups (but then I am a biased geometric group theorist).

EDIT: David Joyner is one of the authors of the book "Applied Abstract Algebra". This looks interesting and very relevant, but I know nothing about it other than what Google can provide.

Pinter's "A Book of Abstract Algebra" (Dover, 2nd edition 2010) covers beginning abstract algebra, carefully setting up illustrative examples.

I think there is no specific field to use in our daily life. However such type of course explored the logical thinking capacity of readers.

• Welcome to the site. Unfortunately, this is not an answer to the question. Some of the other answers to the question might be good for you to read! – Chris Cunningham Mar 23 '18 at 18:55