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When seeing groups and such for the first time, the abstraction often seems pointless and unnecessary to students. (Most students at my school leave their introductory abstract algebra class thinking it was pointless and unnecessary, too.)

What are some nice questions that can be used during the class to motivate the development of groups (or rings), that groups solve nicely? I'd rather Galois theory wasn't used as an example - it takes a bit too much work and wouldn't fit into the type of introductory course I'm thinking of.

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  • $\begingroup$ You might find some interesting things in an MO question on impressing students with group theory mathoverflow.net/questions/13320/… $\endgroup$ – quid Mar 17 '14 at 22:48
  • $\begingroup$ How long would you be willing to spend on an example? Seeing the hyperbolic metric arise as the unique metric preserved by the biholomorphisms of the disk is a great story tying together complex analysis, riemannian geometry, and group theory, but it would probably take a couple sessions to make sense of it all. $\endgroup$ – Steven Gubkin Mar 17 '14 at 22:53
  • $\begingroup$ @StevenGubkin The idea would be to incorporate it into the class itself; make solving the problem a story that's unraveled as the class goes on. Your suggestion would be great if the intro course I'm thinking of wasn't so early in the students' mathematical career (taught at my institution immediately after the 'intro to proofs' course) $\endgroup$ – user37 Mar 18 '14 at 1:14
  • $\begingroup$ @BrianRushton Yes, that answers my question. Since this is a duplicate, should I delete the question, or just vote to close? $\endgroup$ – user37 Mar 18 '14 at 17:38
  • $\begingroup$ @Typically, you vote to close. The way that your question is worded is still valuable, and serves as a `signpost' to direct others to the other question. Also, you still retain the reputation you received from this question. $\endgroup$ – Brian Rushton Mar 18 '14 at 17:44
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I have never tried this, but I think solving a rubik's cube would be a fun application. Try:

http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik's%20Cube.pdf

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