 Take any group formula_1 of order formula_2.Denote by formula_3 the group formula_4
of permutations of elements of formula_1, by
formula_6 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 formula_10
is Hall's universal group "U".