# Ideas for group theory for high school math project

• malawi_glenn
In summary: No he has not.I would recommend something with the Dihedral Groups, but no illustrations/visualizations criteria rules out.

#### malawi_glenn

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Hi

As high school teacher, I sometimes have those extremely talanted and self driven pupils.
In their final year, they are required to make a science or math project, roughly one month full-time studies, approx 15-20 pages report.
This academic year, one of my students have learned some group theory as a "for fun summer project" and was asking me today for suitable projects with this topic.

Now I have no idea how to reply, since afaik everything that is interesting to know can be found online or in textbooks.

It has to be a "scientific" project, so illustrations, visualizations etc are kinda off-limits.

Do any of you guys have any ideas for suitable projects?

I have an unpublished idea about Lie algebras.

It is hard to find something new in group theory. Maybe something in crystallography, or the discrete Fourier transformation for error correcting codes.

malawi_glenn
Not sure student is ready for Lie Algebras, we are talking about high school, 18y old kid.

I also had something in mind along crystallography and point groups.

Cyosis
malawi_glenn said:
Not sure student is ready for Lie Algebras, we are talking about high school, 18y old kid.

I also had something in mind along crystallography and point groups.
Yes, but Lie algebras are even easier than groups and require the same algebraic mindset. There is no need to talk about Lie groups and calculus. Vector space, Leibniz rule, and cross product or matrix multiplication as an example are enough.

My idea is that ##\mathfrak{A(g)}=\{\alpha \,|\,[\alpha (X),Y]+[X,\alpha (Y)]=0\}## defines again a Lie algebra and ##\mathfrak{g}##-module via ##(X.\alpha)(Y) =[adX,\alpha ](Y)=[X,\alpha (Y)]-\alpha ([X,Y]),## i.e. a representation. ##\mathfrak{A(g)}=\{0\}## for simple Lie algebras and ##\mathfrak{A(g)}\neq\{0\}## for solvable Lie algebras (Lie's theorem).

See, only linear algebra required.

vanhees71
Problem is, this student has not taken my "linear algebra for high school students" (yet)

swampwiz
malawi_glenn said:
Problem is, this student has not taken my "linear algebra for high school students" (yet)
... in which case I vote for cryptography and error correcting codes. Maybe simple ones. The discrete FFT can become troublesome.

vela
What do you guys think about coloring of faces on a cube?

malawi_glenn said:
What do you guys think about coloring of faces on a cube?
I think coding theory is more up-to-date and provides more to learn.

But I am out here. I am notoriously bad with coordinates or counting. And coloring cubes sounds like a lot of counting, rotating, and mirroring. If it should be a geometric problem, how about wallpaper groups?

malawi_glenn
You could also have him look the mathematics behind how public-key cryptography works.

malawi_glenn said:
Hi

As high school teacher, I sometimes have those extremely talanted and self driven pupils.
In their final year, they are required to make a science or math project, roughly one month full-time studies, approx 15-20 pages report.
This academic year, one of my students have learned some group theory as a "for fun summer project" and was asking me today for suitable projects with this topic.

Now I have no idea how to reply, since afaik everything that is interesting to know can be found online or in textbooks.

It has to be a "scientific" project, so illustrations, visualizations etc are kinda off-limits.

Do any of you guys have any ideas for suitable projects?
The coloring faces on a cube is neat.
Has the student learned modular arithmetic? If so, RSA Encryption may be doable. I can scan pages from a book which explains it in a simple manner, and send them to you. So that you may judge if it is indeed doable or not.

I would recommend something with the Dihedral Groups, but no illustrations/visualizations criteria rules out.

The first thing coming to my mind is using group theory to solve Rubic's cube.

haushofer said:
The first thing coming to my mind is using group theory to solve Rubic's cube.
I have seen a paper about it. I think this is heavy stuff at the high school level.

MidgetDwarf said:
Has the student learned modular arithmetic? If so, RSA Encryption may be doable. I can scan pages from a book which explains it in a simple manner, and send them to you. So that you may judge if it is indeed doable or not.
No he has not.
MidgetDwarf said:
I would recommend something with the Dihedral Groups, but no illustrations/visualizations criteria rules out.
What I meant was that illustrations/visualizations can not be the goal of the project.
haushofer said:
The first thing coming to my mind is using group theory to solve Rubic's cube.
As mentioned, it is "too heavy". Also, it is just a matter of googling.

fresh_42 said:
I have seen a paper about it. I think this is heavy stuff at the high school level.
I think it's perhaps the most visual and approachable introduction to a heavy stuff subject ;)

malawi_glenn said:
...
In their final year, they are required to make a science or math project, roughly one month full-time studies, approx 15-20 pages report.
...
Now I have no idea how to reply, since afaik everything that is interesting to know can be found online or in textbooks.
...
So how does it work usually for these projects? Can you give us examples of previous projects to get an idea what they are like and how they are designed to avoid what is found online and in textbooks?

malawi_glenn said:
No he has not.

What I meant was that illustrations/visualizations can not be the goal of the project.

As mentioned, it is "too heavy". Also, it is just a matter of googling.

I don't really get this part, at the high school level isn't every project going to be able to be solved by just googling?

Is the real problem here that most projects do some experiment that is like, maybe technically novel, but you can't that in math?

vanhees71
martinbn said:
So how does it work usually for these projects? Can you give us examples of previous projects to get an idea what they are like and how they are designed to avoid what is found online and in textbooks?
Very seldom students do it in maths, and then it is usually like playing around with / applying differential equations to say population dynamics etc. Another popular route for students at my school is to do something related to linear algebra, at least those who have or are taking "my" linear algebra for high school students course. However, as I mentioned earlier, this student has not taken or is taking that course.

For this project course, physics, chemistry and biology is way more popular since it is quite easy there to formulate a problem, what parameters to vary and conduct an experiment and do data analysis.
Office_Shredder said:
Is the real problem here that most projects do some experiment that is like, maybe technically novel, but you can't that in math?
Exactly, see above.

Another idea I had was to let the student see if there is any "project Euler" problems that can be analyzed with group theory, but I do not have that time myself to check em. But then it could also just be a matter of googling and the problem statement would almost be too narrow.

I did find some ideas here https://npflueger.people.amherst.edu/350-18fall/handouts/topics.pdf but perhaps those will be too advanced for him.

At first I thought this would be impossible. But I tried googling and found this

This actually feels like it would have exactly the flow you would want for a project, except that I already found it off Google. Perhaps there is a similar question that they could investigate with the help of a computer, but I don't know what it is.

malawi_glenn
@Office_Shredder that looks ok tbh, have to ask student if he knows basic programming though. .

haushofer said:
I think it's perhaps the most visual and approachable introduction to a heavy stuff subject ;)
The Rubik's cube group is really laaarge. IIRC then the paper I saw has been a master thesis.

If it has to be something new, then nilpotent or solvable groups are a good start. They are far from being classified. On the other hand, it is difficult to decide what is actually new.

malawi_glenn
Have you considered broadening the assignment to include expository papers? The student could study group theory in depth (at a high school level), its motivating concepts and modern developments, and then express what they learned in a way that hopefully justifies their interest to others. He or she or they might also consider learning representation theory and exploring its connections to quantum mechanics: that field has a somewhat high learning curve (most of the work is done by theoretical physicists, who aren't especially notorious for their mathematical clarity), but a serious attempt at an expository paper could make for a good head start. Also, it might be worth keeping in mind that many fields of modern math (except perhaps for combinatorics, which tends to be somewhat accessible and broad) have a somewhat high conceptual barrier to entry because they rely implicitly on familiarity with an array of subjects that are fairly sophisticated (e.g. commutative algebra, algebraic geometry, measure theory, etc.) In fact, there are some who would argue that a math student should prioritize learning fundamentals and only start research when they are inspired to do so (and certainly not on demand.) Unless the "math" projects are mostly about applying ideas from machine learning, tensor/artificial neural networks (fitting functions to data), which is fairly accessible, it might be best to emphasize learning/exploration over novelty.

Couchyam said:
Have you considered broadening the assignment to include expository papers?

The syllabus for this project/assignment is dictaded by the school law in my country. Therefore, expository papers are kinda off-limits.

Another possibility is research instead of invention. Any historical essay (as long as it is not copied) is new in a scientific sense and doable for high school students. The history of <...> can be very motivating! Possible plug-ins:
- group theory (work to do)
- the classification of finite simple groups (possibly difficult)
- Lie groups (I think it started with Lie and Noether and the who-is-who for physicists: Christoffel, Ricci, Riemann, Lagrange, Lipschitz, Lorentz, Weyl, Klein acc. to Noether)
- linear algebraic groups (more algebra than calculus)
- finite groups

malawi_glenn said:
The syllabus for this project/assignment is dictaded by the school law in my country. Therefore, expository papers are kinda off-limits.
I'm sorry to hear that. Have mathematics educators in your country considered changing that law to broaden the spectrum of learning pathways?

PhDeezNutz
malawi_glenn said:
@Office_Shredder that looks ok tbh, have to ask student if he knows basic programming though. .

Random idea, maybe you could investigate the average order of an element of a group of order n, and see if there are any interesting results numerically based on how it factors into primes. I have no idea what result you might get.

malawi_glenn
That sounds like a lot of work to do and demands a restriction on n.

I remember that I had a lot of group theory questions in the challenge threads. Searching for "group" gave me 334 hits. "Group theory" has 10 hits. It could give you ideas (and an overview of standard results which are already known) to check the solution manual (last attachment)

fresh_42 said:
That sounds like a lot of work to do and demands a restriction on n.
Well it does not have to be a conclusive work.
fresh_42 said:
I remember that I had a lot of group theory questions in the challenge threads. Searching for "group" gave me 334 hits
I will check it out.

malawi_glenn said:
Well it does not have to be a conclusive work.
Yes, but "average" is hard if you do not even know the denominator, how many groups of a given order.

Fresh, your average is over too large a space. Take a specific group of order n, compute the average order of an element in that group. Do this for whatever groups you are able to construct and see if you notice anything

Edit to add: as an example of a question you could ask, is the cyclic group of order n the group of order n with the highest average order element? A counterexample would be cool (I have no idea is it exists), a hypothesis of when it's true or false based on any counterexamples you find sounds like it would make a good project, even if you don't prove anything mathematically?

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malawi_glenn and fresh_42
Office_Shredder said:
Fresh, your average is over too large a space. Take a specific group of order n, compute the average order of an element in that group. Do this for whatever groups you are able to construct and see if you notice anything

Edit to add: as an example of a question you could ask, is the cyclic group of order n the group of order n with the highest average order element? A counterexample would be cool (I have no idea is it exists), a hypothesis of when it's true or false based on any counterexamples you find sounds like it would make a good project, even if you don't prove anything mathematically?
Sounds like a cool scoope for a project at this level.

malawi_glenn said:
Problem is, this student has not taken my "linear algebra for high school students" (yet)
The very bare understanding I have of abstract algebra is solely from having deep knowledge of linear algebra. Unless this high school is a super-duper magnet school, I just don't see how anyone will have type of mathematical sophistication to do anything in this.

swampwiz said:
The very bare understanding I have of abstract algebra is solely from having deep knowledge of linear algebra. Unless this high school is a super-duper magnet school, I just don't see how anyone will have type of mathematical sophistication to do anything in this.
No idea what a magnet school is.
But yes linear algebra is very good to know. But you can do the entire discrete mathematics by Biggs without knowing any linear algebra.

You'd be suprised what some of my students can ro at the age of 17-18. Some of their project work is more sophisticated than most bachelor theses...

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vanhees71 and fresh_42
malawi_glenn said:
You'd be suprised what some of my students can ro at the age of 17-18. Some of their project work is more sophisticated than most bachelor theses...
I like this point of view. There is no such thing as being too young to understand. It is only a matter of language, maybe of time, but not a matter of content. I will never forget, and I am absolutely sure she has forgotten, how my then-daughter of six years of age could handle negative numbers, including some basic arithmetic in ##\mathbb{Z}[ i ].## Of course, it was more about playing games against boring car rides. However, it shows what humans are capable of without the pressure of grades or the narrow - and in my mind silly - corset of a curriculum.

vanhees71 and berkeman
swampwiz said:
The very bare understanding I have of abstract algebra is solely from having deep knowledge of linear algebra. Unless this high school is a super-duper magnet school, I just don't see how anyone will have type of mathematical sophistication to do anything in this.
I have seen a few kids over the years that are extremely brilliant, and new able to handle pure mathematics. the school I got my bs from has an early entrance exam. Where the majority of students are between 16-17 (close to 18). Currently, there is a 14 year old student taking ring theory, and should graduate next year. But yes, it is not the common experience of most high school students.

When I taught calculus to high schoolers, I actually introduced a bit of analysis, and they were fine with it. Like others mentioned, language, presentation, and teacher support or what is needed, and not caring what administration says.

OK, but the important question here is, how did the project go? What did they end up doing?