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

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

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

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