Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

Abstract

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.

On the Genus of the Commuting Graphs of Finite Non-Abelian Groups

“…However, in [11], it is shown that, for all positive integers d, there exists a finite special 2-group G such that the commuting graph of G has diameter greater than d. But in [17], it is proved that for finite groups with trivial center the conjecture made in [14] holds good. The concept of commuting graphs of groups (taking, as the vertices, the non-trivial elements of the group in place of non-central elements) has also been recently used in [20] to show that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. There is also a ring theoretic version of commuting graphs (see, for example, [2], [3]).…”

Investigation of Commuting Graphs for Elements of Order 3 in Certain Leech Lattice Groups

“…They were eminent for giving evidence of a prescribed isomorphism of an involution centralizer, where there is a limited number of non-abelian groups capable of containing it. These graphs were extremely vital for the works of the Margulis-Platanov conjecture [2], as the graphs mentioned in [1] have X = G\{1} where 1 is the identity element of G). When X is a conjugacy class of involution (conjugacy class X of G means that for any two elements x, y  X there are an element ISSN: 0067-2904 g G such that x g =y, whereas involution means that all elements of X have order 2), then the commuting graph is known as the commuting involution graph.…”

Commuting Involution Graphs for 4-Dimensional Projective Symplectic Groups

“…In effect commuting graphs first appeared in the paper of Brauer and Fowler [14], famous for containing a proof that up to isomorphism only finitely many nonabelian simple groups can have a given centralizer of an involution. The commuting graphs considered in [14] had X ¼ Gnf1g-such graphs have played an important role in recent work related to the Margulis-Platanov conjecture (see [41]). Various kinds of commuting graphs have been deployed in the study of finite groups, particularly the non-abelian simple groups.…”