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Section 1.12 Fun stuff

Let me end this section on finite groups with three interesting problems to think about.

  1. Can you prove that the binomial coefficient

    \begin{equation*} \begin{pmatrix} m+n \\ m \end{pmatrix} = \frac{(m+n)!}{m! n!} \end{equation*}

    is always an integer for any positive integers \(m,n \in \mathbb{Z}\) using group theory?

  2. The set of sensible orientations of a rectangular mattress on a bed forms a group. What group is it? What strategy can you take to rotate periodically your mattress between all its orientations?

  3. For more fun stuff have a look at the Futurama theorem... It's really fun! See for instance https://en.wikipedia.org/wiki/The_Prisoner_of_Benda and https://theinfosphere.org/Futurama_theorem, as well as https://arxiv.org/abs/1608.04809 for an interesting generalization.

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