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How to introduce Group Theory to a general audience in 15 minutes?

I know that it will be quite tough to introduce Groups to a general audience in such a short time.

So what will be a good way to introduce Groups using Chalk-Duster approach?

What are the topics in Group Theory which can be taught in 15 mins

Any help.

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    $\begingroup$ What is a "Chalk-Duster approach"? $\endgroup$ – Joel Reyes Noche Dec 10 '18 at 4:04
  • $\begingroup$ Also, you tagged your question "undergraduate-education." Are you saying that your audience is composed of undergraduate students? Or are they teachers of undergraduates? $\endgroup$ – Joel Reyes Noche Dec 10 '18 at 4:06
  • $\begingroup$ Doing it using chalk no powerpoint $\endgroup$ – user10666 Dec 10 '18 at 4:23
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    $\begingroup$ It means they are undergraduate students and also teachers who teach undergraduate students $\endgroup$ – user10666 Dec 10 '18 at 4:23
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One way to introduce group theory to a general audience is to talk about how to flip a mattress so that it is in "a different position each time, so as to pound down the lumps and fill in the sags on all the various surfaces."

Brian Hayes has a good discussion of this in his September-October 2005 Computing Science column in American Scientist. You'll probably need to make a drawing (on a blackboard) like the one below (which I took from his article).

Klein 4-group

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Well, it depends a lot on what your real goal is. If it's just the wow factor, you could introduce the Rubik's Cube group or do computations about it. If they know what a matrix is, then doing matrix computations would be a very nice way (I mean $GL_n(\mathbb{R})$ here, not additive matrices). Or symmetries in art/frieze groups.

However, my personal favorite (even though the groups are commutative) is simply introducing clock arithmetic or one's digit arithmetic (which you may know as $\mathbb{Z}_{12}$, $\mathbb{Z}_{10}$, or as $\mathbb{Z}/12\mathbb{Z}$, $\mathbb{Z}/10\mathbb{Z}$ if you are a notational purist). Just a little messing around with these can be quite powerful if you have people truly new to the concepts. And the huge benefit is that everyone is very familiar with them, even if they didn't know it. Naturally this is for a less mathematically inclined audience than some others might be. But definitely can do everything in 15 minutes.

(Also: Want multiplication but there are zero-divisors? No problem, just ignore them and now you have another, "multiplicative", group of units. Want a non-commutative variant? Finish up by talking about clock arithmetic as a rotation of the clock face - and ask how flips would work.)

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I like the mattress example. I was going to say 3d symmetry point groups (since it is visual and useful plus huge impact in chemistry and physics). But...

The Reyes answer does what I want and makes it even simpler. I would really just cover that example, maybe even make some 3x5 cards with a lump and let audience play with them. Covering that one example and the table of results is fine. And the insight that two motions can equal a third one is COOL INSIGHT. That will get them a good feeling.

On top of that just give a generic discussion of group theory to answer basic questions: what course is it taught in (what year) for math majors, when chemists/physicists cover it. and that it has uses in chem/physics (spectroscopy, molecule ID, etc.) as well as "proving you can't solve 5th order polynomials the way you do with quadratic).

DO NOT show anything like wierd Z letter thing from kccrisman answer. NO, NO, NO. NO matrices, group names. NONE OF THAT. You don't have time to cover wierd notation in 15 minutes. Just do the one mattress example and then give them orienting information about the topic AS A TOPIC (not proving anything).

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    $\begingroup$ Your answer seems to respond to other users' answers instead the main question. I suggest making this instead into two comments, on those other answers, respectively. $\endgroup$ – Brendan W. Sullivan Dec 11 '18 at 18:19
  • $\begingroup$ I feel ya. There's a lot of comment stuff, but it would be a mess to sort it into two comments and you lose some of the insights (e.g. about topic orientation or even general warning against mathy notation...not just versus what Kris suggested but as a general warning, worthy of larger font.) It;s a dog's breakfast but I would leave it as is. $\endgroup$ – guest Dec 11 '18 at 18:23

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