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.

It was defined by Philip Hall in 1959,[1] and has the universal property that all countable locally finite groups embed into it.

Hall's universal group is the Fraïssé limit of the class of all finite groups.

Construction

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Take any group   of order  . Denote by   the group   of permutations of elements of  , by   the group

 

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

 

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

 

A direct limit (that is, a union) of all   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  . The group   acts on   by permutations, and conjugates all possible embeddings  .[1]

References

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