String duality

In algebra, Hall's universal group is a countable locally finite group, say U, which is uniquely characterized by the following properties.

• Every finite group G admits a monomorphism to U.

• All such monomorphisms are conjugate by inner automorphisms of U.

It was defined by Philip Hall in 1959.

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 $G\hookrightarrow U$.