K0 of Hallʹs universal group

We consider the limiting behaviour of the character theory of finite groups. More formally, we consider the Grothendieck group X(U) of finitely generated projective modules over the complex group algebra of Hallʹs universal countable locally finite group U. This group contains a copy of every finite group and two isomorphic finite subgroups are conjugate. We note that X(U) is a commutative ring and describe its structure explicitly.