We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let
D
D
be a finite dimensional division algebra having center
K
K
, and let
N
⊆
D
×
N\subseteq D^{\times }
be a normal subgroup of finite index. Suppose
D
×
/
N
D^{\times }/N
is not solvable. Then we may assume that
H
:=
D
×
/
N
H:=D^{\times }/N
is a minimal nonsolvable group (MNS group for short), i.e. a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote Property
(
3
1
2
)
(3\frac {1}{2})
. This property includes the requirement that the diameter of the commuting graph should be
≥
3
\ge 3
, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of
D
×
/
N
D^{\times }/N
has Property
(
3
1
2
)
(3\frac {1}{2})
, then
N
N
is open with respect to a nontrivial height one valuation of
D
D
(assuming without loss of generality, as we may, that
K
K
is finitely generated). After establishing the openness of
N
N
(when
D
×
/
N
D^{\times }/N
is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of
K
K
over its prime subfield to eliminate
H
H
as a possible quotient of
D
×
D^{\times }
, thereby obtaining a contradiction and proving our main result.