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The question What are some good mathematical applications to present in an abstract algebra course? asks about mathematical applications of abstract algebra.

What are some applications of abstract algebra outside of mathematics? These could be included either in a traditional abstract algebra course or in an applied algebra course. Which abstract algebra textbooks have particularly good support for applications or have a focus on applied algebra?

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    $\begingroup$ You might look up “point groups” in physical chemistry. $\endgroup$
    – Steve
    Oct 9, 2020 at 13:53
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    $\begingroup$ I imagine there will be several answers with specifics, but if not and you're looking for something to mention (or possibly even include) when teaching, you can simply google something like "group theory" (probably holds much more promise than anything else I can think of right now) along with various scientific fields/topics, such as this. Off the top of my head I know that most physical chemistry texts and quantum chemistry texts have chapters on group theory. (While I was writing my comment @Steve gave this example.) $\endgroup$ Oct 9, 2020 at 13:57
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    $\begingroup$ The calculations in algebraic topology boil down to algebra after a while. I believe modern cutting edge quantum mechanics now involves such topology and hence such algebra. More classically, tesselations and finding shapes which are invariant under transformations leads to designs which are more or less applied abstract algebra. Of course such designs may precede the algebra, but they are better understood in the light of the algebra. Then, representation theory in physics is applied abstract algebra, all the "quantum numbers" are born of representation theory which is abstract algebra... $\endgroup$ Oct 9, 2020 at 15:38
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    $\begingroup$ Feel free to post comments like the above as answers. $\endgroup$
    – J W
    Oct 9, 2020 at 16:54
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    $\begingroup$ My answer to the linked question listed applications in telecommunication engineering. Coding theory and cryptography in particular are full of applied algebra. But, I cannot recommend a textbook covering those themes in a book dedicated to algebra. Sure, some of the applications are widely known, but not many authors of books on algebra are familiar with too many of them. My coverage (when given a chance) is based on first hand experience rather than textbooks. $\endgroup$ Oct 9, 2020 at 20:24

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There are a lot of applications of group theory in chemistry and physics. (I know the chemistry side of it better.)

Point groups of molecules describe their symmetry and thus their spectroscopy (bending modes for IR spectra). See for instance,

https://www.amazon.com/Chemical-Applications-Group-Theory-3rd/dp/0471510947

OR

https://www.youtube.com/watch?v=BjknQc3bQ8o (note the character tables)

OR

https://www.youtube.com/watch?v=pEw0RKCANFs (simple visual)

Space groups are intrinsic to X-ray crystallography, which describes a huge amount of matter (even things you don't think of as crystals, like metals or ceramics or [bulk of] computer chips, are crystalline solids). See for instance

https://it.iucr.org/

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Thanks in part to Jyrki Lahtonen's remarks on coding theory and cryptography, I have an applied algebra book to suggest:

Algebra for Applications, Arkadii Slinko, 2nd ed., 2020 Springer

The preface to the first edition states:

This book originated from my lecture notes for the one-semester course which I have given many times in The University of Auckland since 1998. The goal of this book is to show the incredible power of algebra and number theory in the real world. It does not advance far in theoretical algebra, theoretical number theory or combinatorics. Instead, we concentrate on concrete objects like groups of points on elliptic curves, polynomial rings and finite fields, study their elementary properties and show their exceptional applicability to various problems in information handling. Among the applications are cryptography, secret sharing, error-correcting, fingerprinting and compression of information.

It does not replace a more traditional course on abstract algebra and indeed at The University of Auckland where Slinko teaches Algebra and Applications, a conventional course, Algebraic Structures, is also taught (see list of undergraduate courses).

For an MAA review of the first edition by Mark Hunacek, see https://www.maa.org/press/maa-reviews/algebra-for-applications. (For a mini review of the second edition mentioning the changes made, see here.)

Whether Slinko's book counts as "a book dedicated to algebra", to borrow Jyrki Lahtonen's phrase, is debatable I suppose. I think Slinko tries to strike a balance between introducing the necessary algebra and giving sufficient space to the applications. I also think that in general, the line is blurred somewhat between algebra, number theory and applications. For example, some searching has revealed Niederreiter & Winterhof's Applied Number Theory, which after a review of some number theory and algebra, covers cryptography, error-correcting codes, quasi-Monte Carlo methods, pseudorandom numbers and miscellaneous further applications. Another example is Hoffstein, Pipher & Silverman's An Introduction to Mathematical Cryptography, which naturally focuses on cryptography, but introduces the necessary topics from number theory, algebra, probability and information theory to keep the book fairly self-contained. The contents of these books and Slinko's book clearly overlap to some extent.

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The answers above are good - crystallography and cryptography are huge. Here are a couple more "smaller" applications that occur to me that show up in textbooks fairly frequently (both group theory, not rings/fields). It would be tough to make a whole course out of either one of these, but definitely doable to spend a day to a week or more covering these sorts of applications if you didn't have to cover ground in preparation for fields/modules/Galois theory.

  • Puzzle solving, such as Rubik's Cube. There is at least one textbook (of sorts) that focuses heavily on this (with Sage code too!) by David Joyner.

  • Pretty much any combinatorics book beyond a very first one will cover things like the (Cauchy-)Frobenius(-Burnside) Lemma for counting various patterns, which is a serious application of group theory (even in music theory). Keller and Trotter is one I have used bits and pieces of on occasion, and it's open source too.

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    $\begingroup$ I second Burnside's lemma as an important tool in combinatorics. One fun puzzle is counting the number of Tsuro tiles. Actually, a lot of board game design combinatorics uses Burnside's lemma. math.stackexchange.com/questions/1270528/… $\endgroup$ Jan 25 at 12:04
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    $\begingroup$ True, though the OP was about things outside of math, which not every combinatorial application is :) $\endgroup$
    – kcrisman
    Jan 25 at 18:48
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A bit of a strange and abstract answer, but for variety:

I have found personally that even just the basics of abstract algebra and group theory have helped me immensely pedagogically, as it has allowed me to understand abstractions better, and fundamentally altered how I think and process ideas and concepts.

Firstly, group theory is essentially the study of symmetry. It helps you understand how things can be symmetrical, and the different ways you can represent the same object. Anywhere that symmetry exists, you can think of it in terms of group theory. Let's say you and 3 other friends are playing a game together, and for variety sake, you want to switch up the player order each game. With a good understanding of the $S_4$ group, you don't need to remember which orders you've tried, you can easily construct many simple ways to determine the next ordering. This example is a little contrived, but hopefully you can see where it could appear innocuously in every day life. Most people won't see "player order" as something having symmetry.

Here is an example to make the next paragraph feel less abstract: Let $a$ be a rotation 90 degrees clockwise around the $x$-axis. Let $b$ be a rotation 90 degrees clockwise around the $y$-axis. Let $c$ be a rotation 90 degrees counter-clockwise around the $x$-axis. I have a cube and I want to apply the $a$, $b$, and $c$ rotations in that order. What will the cube look like?

First we represent the overall rotation as $a+b+c = a+c+b$. The $a$ and $c$ rotations cancel, so we are just left with $b$. Grab a cube and try it yourself! You will find this does not get you the right answer. Wait, what?

When learning math, there are several major hierarchies of concepts students learn as they get more advanced, in the sense that they must be very comfortable manipulating and understanding them in order to reach the next step of mathematical maturity: numbers, arithmetic, expressions, functions, logic, properties. The "properties" level is examined when you study abstract algebra. What do algebras look like if the commutative and associative properties don't hold? There are a lot of implicit assumptions we make when writing mathematical statements that we often take for granted, that are now muscle memory (like when we solve an equation). But by examining the properties of the algebra, we can catch our mistakes and justify our reasoning. In this problem, we assumed that 3D rotations are commutative, that $a+b = b+a$, but this is in fact not true (this is a special case of matrix multiplication). Having a good grasp of abstract algebra makes it easier for you to challenge those assumptions.

In general, the idea that you can basically do algebra with just about anything as long as your operations are defined properly, is a pretty cool and powerful idea. The idea that the algebra systems are (sort of) defined by their properties and not by what the elements represent means seemingly unrelated things are the same. This changes one's attitude towards math significantly, and you start to realize there is just as much invention in math as there is discovery. We aren't just proving theorems, we are also inventing definitions.

Once you broaden your perspective in this way, you will have a healthier approach towards higher mathematics, a shift in philosophy, and an appreciation for abstraction. Then it doesn't seem too difficult to imagine that there could be plenty of applications of these concepts intuitively through art and design, programming, playing games, etc. even if it is difficult to identify specific ones. The more complex the math, the less explicit the applications and the more important they are in changing how we think and our philosophies.

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This is a classic article:

Hayes, Brian. Group Theory in the Bedroom, and other Mathematical Diversions. Hill and Wang, 2008. Book link. "Having run out of sheep the other night, I found myself counting the ways to flip a mattress. ..."

           

(The title refers to one article. The entire collection is worthwhile.)

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For the group theory in physics, the following books looks very promising:

Group Theory in a Nutshell for Physicists by Anthony Zee

(It just came out in 2022 and I haven't gotten a chance to read it carefully, but the reviews are very good).

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