TITLE: How powerful is Cayley's theorem?
QUESTION [19 upvotes]: So the Cayley's theorem gives a subgroup $H$ of $S_n$ for a $G$ such that $G$ is isomorphic to $H$. So $S_n$ behaves like a Universal set to $G.$
Is there  a smaller universal object for all groups of size $n$?

REPLY [7 votes]: As a first example of such a "smaller universe", think of any finite $G$ having a proper normal subgroup, $H$. Then, $G$ acts by conjugation on $H$, and the kernel of this action is $C_G(H)$. Therefore, if $C_G(H)$ is trivial, then $G$ embeds into $S_{|H|}$, which is a "smaller universe" than (Cayley's) $S_{|G|}$. For instance, $C_{S_4}(A_4)=\{1\}$ and hence $S_4\hookrightarrow S_{12}$ (with no claim that this is the minimal embedding).
On the contrary, if $|G| = p > m$ is prime, then $p \not\mid |S_m| = m!$. Therefore there cannot be an embedding $G \hookrightarrow S_m$, because its image would be a subgroup of order $p$, contradicting Lagrange's theorem. So, in this case, Cayley's one is the "smallest possible universe".