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 ${\mathrm{\Gamma }}_0$ of order $\ge 3$. Denote by ${\mathrm{\Gamma }}_1$ the group $S_{{\mathrm{\Gamma }}_0}$ of permutations of elements of ${\mathrm{\Gamma }}_0$, by ${\mathrm{\Gamma }}_2$ the group

$S_{{\mathrm{\Gamma }}_1}=S_{S_{{\mathrm{\Gamma }}_0}}$

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

${\mathrm{\Gamma }}_0\hookrightarrow {\mathrm{\Gamma }}_1\hookrightarrow {\mathrm{\Gamma }}_2\hookrightarrow \cdots .$

A direct limit (that is, a union) of all ${\mathrm{\Gamma }}_i$ 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 ${\mathrm{\Gamma }}_i\subset U$. The group ${\mathrm{\Gamma }}_{i+1}=S_{{\mathrm{\Gamma }}_i}$ acts on ${\mathrm{\Gamma }}_i$ by permutations, and conjugates all possible embeddings $G\hookrightarrow U$.