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.
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
by permutations, and conjugates all possible embeddings