Is it possible to realize any finite group as a subgroup of a semisimple Lie group?

Question

This question seems rather trivial after seeing the answers. So now I would also like to know what's the lowest dimension of the Lie group given the rank or the order of the finite group? For example, for dihedral groups, no matter how large the order is, we only need a one dimensional Lie group.

Answer 1

By Cayley, every finite group is a subgroup of $S_n\subseteq GL_n(\mathbb{R})$. Here $S_n$ is represented by permutation matrices. This has been proved at this question in detail:

Show that any finite group $G$ is isomorphic to a subgroup of $GL_{n}(\mathbb{R})$ for some $n$.

Answer 2

You can also realize $S_n$ inside $SL(n+1, \mathbb{R})$ (indeed inside $SO(n+1, \mathbb{R})$ by the map $$\sigma\mapsto (P_{\sigma}, \epsilon(\sigma) )$$