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, 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 . References Infinite group theory Permutation groups