The Margulis-Platonov conjecture describes noncentral normal subgroups of the rational-point group of
an absolutely almost simple simply connected algebraic group over a global
field by finitely many compact local groups. For a global
field 
and such a group 
, let
 |
where 
is the completion at the place 
and the rank is the dimension
of a maximal 
-split algebraic torus.
The set 
is finite, and 
is a compact group with its product
topology. Write 
for the diagonal homomorphism. The assertion
is that every normal subgroup 
not contained in the center of 
has the form
 |
for an open normal subgroup 
of 
. No topological closedness or finite-generation
hypothesis is imposed on 
. If 
is empty, the conclusion is 
.
