Hall's universal group

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,^[1] and has the universal property that all countable locally finite groups embed into it.

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 $G \hookrightarrow U$.

References

1. Hall, P. Some constructions for locally finite groups. J. London Math. Soc. 34 (1959) 305--319. MR 162845