  In 1955 Dye proved that two von Neumann factors not of type I_2n are
isomorphic (via a linear or a conjugate linear *-isomorphism) if and only if
their unitary groups are isomorphic as abstract groups. We consider an analogue
for C*-algebras. We show that the topological general linear group is a
classifying invariant for simple, unital AH-algebras of slow dimension growth
and of real rank zero, and the abstract general linear group is a classifying
invariant for unital Kirchberg algebras in the UCT class.
