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Problem 32 (Relationship Between S And Its Span)

Let $S$ be a set of permutations of $X$.

  1. Prove that $S\subseteq \text{span}(S)$. In other words, we know that $S$ is always contained in its span; the span might be larger.
  2. Prove that if $\text{span}(S)\subseteq S$, then $S$ is closed.
  3. If $T\subseteq S$, then show that $\text{span}(T)\subseteq\text{span}(S)$.

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