  In the theory of unitary group representations, a group is called type I if
all factor representations are of type I, and by a celebrated theorem of James
Glimm [Gli61b], the type I groups are precisely those groups for which the
irreducible unitary representations are what descriptive set theorists now call
"concretely classifiable". Elmar Thoma [Tho64] proved the following surprising
characterization of the countable discrete groups of type I: They are precisely
those that contain a finite index abelian subgroup. In this paper we give a
new, simpler proof of Thoma's theorem, which relies only on relatively
elementary methods.
  [Gli61b] James Glimm, Type I $C^{\ast} $-algebras, Ann. of Math. (2) 73
(1961), 572--612. MR 0124756
  [Tho64] Elmar Thoma, \"Uber unit\"are Darstellungen abz\"ahlbarer, diskreter
Gruppen, Math. Ann. 153 (1964), 111--138. MR 0160118
