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Problem 40 (Properties Of Closed Sets Of Permutations)

Let $H$ be a nonempty closed set of permutations of a set $X$.

  1. Show that the identity function $id_X$ is in $H$.
  2. Show that if $\sigma\in H$, then so is $\sigma^{-1}$.
  3. Show that if $\alpha\in H$ and $\beta\in H$, then $\alpha\circ \beta\in H$.
  4. Show that if $\alpha,\beta,\gamma\in H$, then $\alpha\circ (\beta\circ \gamma) = (\alpha\circ \beta)\circ \gamma$.

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