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We just finished a lesson about determinants and the symmetric group with all what comes with it ( permutations, transpositions etc... ), except we didn't do group theory ( we only see it next year ), just some general algebra. But I'd like to know if there are some nice problems online about permutations ( like for example Muirhead's inequality ).

Thanks !

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"Nice" is such a personal thing...and without grout theory that's even more limited. Also I'm not sure how Muirhead's ineq. fits in here... –  DonAntonio May 21 '12 at 11:21
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Grout theory? Well, I guess you can apply it to tilings.... –  Gerry Myerson May 21 '12 at 12:50
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I don't know if they are available on-line, but John D. Dixon's Problems in Group Theory (Dover) Chapter 2 is about permutation groups, and de Souza and Silva's Berkeley Problems in Mathematics, section 6.5, is about $S_n$, $A_n$, $D_n$, etc. –  Arturo Magidin May 21 '12 at 16:39

2 Answers 2

Your question is very general and you should probably specify what you are looking for exactly. One thing I always like to do when students learn about the symmetric group for the first time is talk about the Futurama episode The prisoner of Benda where they ask a question about permutations and even provide an explicit, mathematically correct proof of the solution.

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And this video: youtube.com/watch?v=8M4dUj7vZJc explains the episode and the proof very well. –  Sam Jones May 21 '12 at 12:20

Let $1 < m < n$. Show that $\langle (1 2 \cdots m), (1 2 \cdots n) \rangle$ has a $3$-cycle. (I was trying to remember a problem that I have done in Isaacs' book. I might have remembered wrong, though I think it's right. But since you want a problem, if the statement is wrong, just tweak the problem and prove it wrong.)

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Personally, this was one of the problems that I think was good to get my hands dirty when it comes to understanding some elementary computations and lemmas of symmetric groups. –  GYC Jan 19 '13 at 3:35

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