Please Login to access more options.

Problem (Characterizing Closed Sets Of Permutations)

Let $H$ be a set of permutations of a set $X$. Suppose that $H$ satisifies the following 4 properties.

  1. The identity function $id_X$ is in $H$.
  2. If $\sigma\in H$, then so is $\sigma^{-1}$.
  3. If $\alpha\in H$ and $\beta\in H$, then so is $\alpha\circ \beta\in H$.
  4. If $\alpha,\beta,\gamma\in H$, then we have $\alpha\circ (\beta\circ \gamma) = (\alpha\circ \beta)\circ \gamma$.

Prove that $H$ is closed.

This problem, together with Properties Of Closed Sets Of Permutations, characterizes precisely when a set of permutations is closed. These 4 properties are the beginnings of group theory.

The following pages link to this page.