Hall's universal group

In algebra, Hall's universal group isa 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. [Hall, P."Some constructions for locally finite groups."J. London Math. Soc. 34 (1959) 305--319. MathSciNet | id = 162845]

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

:$xmapsto\; gx$

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

:$Gamma\_0\; hookrightarrow\; Gamma\_1\; hookrightarrow\; Gamma\_2\; hookrightarrow\; ....$

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 anygroup 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