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Problem 45 (Spans Of Permutations Are Subgroups)

Let $X=\{1,2,3,\ldots,n\}$. Let $S_n$ be the set of all permutations of $X$ (so we know there are $n!$ such permutations).

  1. Show that $S_n$ is a group under function composition $\circ$.
  2. If $H$ is a subset of $S_n$ that is closed (under composition combinations of permutations), prove that $H$ is a group under function composition.
  3. Let $S$ be a nonempty subset of $S_n$. Show that $\text{span}(S)$ is a group under function composition.

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