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 ${\displaystyle \Gamma _{0}}$ of order ${\displaystyle \geq 3}$. Denote by ${\displaystyle \Gamma _{1}}$ the group ${\displaystyle S_{\Gamma _{0}}}$ of permutations of elements of ${\displaystyle \Gamma _{0}}$, by ${\displaystyle \Gamma _{2}}$ the group

${\displaystyle S_{\Gamma _{1}}=S_{S_{\Gamma _{0}}}\,}$

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

${\displaystyle x\mapsto gx\,}$

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

${\displaystyle \Gamma _{0}\hookrightarrow \Gamma _{1}\hookrightarrow \Gamma _{2}\hookrightarrow \cdots .\,}$

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

References

1. Hall, P. Some constructions for locally finite groups. J. London Math. Soc. 34 (1959) 305--319. MR162845