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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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.)
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.