I have a question about the symmetric group. Taking signatures of permutations defines a surjective homomorphism $S_n \rightarrow \mathbb{Z}/2$. This is compatible with the natural inclusions $S_n \hookrightarrow S_{n+1}$, so we get a surjection $S_{\infty} \rightarrow \mathbb{z}/2$. Here $S_{\infty} S_{\infty}$ is the direct limit of the $S_n$. In other words, $S_{\infty} S_{\infty}$ is the group of finitely supported permutations of a countable set.