# Group theory for high schoolers, want the opinion of other educators

So I am going to be teaching the basics of group theory to high schoolers in a few weeks, and I want to hear what the Stack Exchange network has to say on the matter.

1. What are the applications and insights I should show them in order to excite them about the subject?
2. What would be a good possible outline of the syllabus, including the rote aspects of group theory, and what are some possible sample problems for the non-rote aspects, i.e. this is why group theory is nifty, these are the deeper connections, these are intriguing applications, etc.?

• I have a lot of experience with teaching group theory to unconventional students - high schoolers, middle schoolers, and middle and high school teachers - but/so the prospect of composing an answer felt overwhelming to me. I invite you to reach out to me by email. – benblumsmith Jul 12 '16 at 7:33
• Also - I once asked a related question: math.stackexchange.com/questions/99339/… – benblumsmith Jul 12 '16 at 7:34
• Some of the links with cryptography are both interesting and fairly accessible. In particular -- Diffie-Hellman makes sense in any group (albeit not secure in every group). The original Diffie-Hellman in the group of units mod p is easy to explain, and elliptic-curve Diffie-Hellman can be discussed in a hand-waving sort of way. – John Coleman Jul 12 '16 at 12:02
• @benblumsmith If you have relevant resources that you could share w/o it being overwhelming -- even if it wasn't a "comprehensive" list -- that would be great! – Benjamin Dickman Jul 12 '16 at 15:12
• Related: A book for abstract algebra with high school level Also of possible interest is my annotated bibliography of "lower level" papers that I give in Exotic Group Examples. – Dave L Renfro Jul 12 '16 at 15:49

Counting problems as application of Burnside theorem usually excites high school students. Even Polya's pattern theorem can be stated without proof and applied to coloring of geometrical shapes.

Here are some applications that may interest students :

• The three great geometrical constructions - Group theory provided impossibility proofs to the three great problems the Greeks couldn't conquer - squaring the circle, trisecting the angle and doubling the cube (all with a compass and unmarked ruler). I would recommend you give your students the task of performing these constructions, and when they can't ask them to come up with impossibility proofs. And, then tell them how a proof for these questions took many centuries and it came with the apparatus of Group Theory. Tell them about the lune of Hippocrates and how he came close to solving the problem of squaring the circle, but didn't quite do it.
• Rubik's cube. That would surely impress a lot of students. Toys like this are very popular with students, and you could explain some facts about how it's a group of permutations.
• I can envision the Rubik's cube, and perhaps things like the lune of Hippocrates (or trisecting with a tomahawk, through origami...) have some marginal interest, but have you seen or enacted lessons around the three classical constructions at the high school level? MESE 9784 asked whether these should even be covered in a Galois Theory course; I would be interested about details regarding their implementation in a group theory intro for pre-college students. Is the "big reveal" that these constructions are impossible a powerful motivator? – Benjamin Dickman Jul 12 '16 at 19:52
• @BenjaminDickman You're right. The proofs might be too complicated. But, I do think the knowledge, of what the subject you're about to study has achieved, is a powerful motivator. – user230452 Jul 13 '16 at 6:12
• You write that the three geometric problems were solved with group theory, but this is not accurate. They were solved with an understanding of polynomials, but none of them rely on Galois theory. How else did you intend group theory to be involved? The label group theory is more specific than the label abstract algebra. – KCd Jul 15 '16 at 12:03

Addressing your 1st question ("excite them about the subject?"), I hope you can make use of Brian Hayes's delightful American Scientist article:

Brian Hayes, Group Theory in the Bedroom. American Scientist, Volume 93, Number 5, 2005, p.395ff. Reprinted in: http://grouptheoryinthebedroom.com/.

Figure 1. The mattress-flipping method recommended by a number of manufacturers and retailers promises to "turn it over and end-to-end," suggesting that the maneuver rotates the mattress around two axes at the same time. In fact the algorithm has the same effect as a single end-over-end flip.