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 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 .