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, 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 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