Hall's universal group

Construction

Take any group \Gamma_0 of order \geq 3 . Denote by \Gamma_1 the group S_{\Gamma_0} of permutations of elements of \Gamma_0 , by \Gamma_2 the group

: S_{\Gamma_1}= S_{S_{\Gamma_0}} ,

and so on. Since a group acts faithfully on itself by permutations

: x\mapsto gx ,

according to Cayley's theorem, this gives a chain of monomorphisms

:\Gamma_0 \hookrightarrow \Gamma_1 \hookrightarrow \Gamma_2 \hookrightarrow \cdots . ,

A direct limit (that is, a union) of all \Gamma_i is Hall's universal group U.

Indeed, U then contains a symmetric group of arbitrarily large order, and any group admits a monomorphism to a group of permutations, as explained above. Let G be a finite group admitting two embeddings to U. Since U is a direct limit and G is finite, the images of these two embeddings belong to \Gamma_i \subset U . The group \Gamma_{i+1}= S_{\Gamma_i} acts on \Gamma_i by permutations, and conjugates all possible embeddings G \hookrightarrow \Gamma_i.

References