Normal subgroups of $GL_n(D)$ are not finitely generated

Abstract

Abstract. As a generalization of Wedderburn's classic theorem, it is shown that the multiplicative group of a noncommutative finite dimensional division algebra cannot be finitely generated. Also, the following conjecture is investigated: An infinite non-central normal subgroup of GLn(D) cannot be finitely generated.

Multiplicative groups of division rings

“…Thereby, he also slightly improved Shirvani's bound for |G : A|. In fact, Wehrfritz results, show that one can replace the bound 60 deg(D) 2 by 30 deg(D) 2 . In the sequel, we try to simplify Shirvani's idea to achieve the major part of the above mentioned results.…”

“…(i) G is either supersoluble or nilpotent; (ii) G has no subgroups isomorphic to SL (2,5) and D has a prime power degree.…”

“…However, a natural question related to (ii) is what happens when G has a subgroup isomorphic to SL (2,5)? Answering this question requires suitable information concerning the structure of soluble-by-finite subgroups of division algebras.…”

On Weakly Locally Finite Division Rings

“…Also, it is known that every positive integer can occur as the Gelfand-Kirillov dimension of some commutative algebra. Moreover, for every real number r ≥ 2, there always exists some k-algebra A with GKdim k (A) = r. Bergman [3] proved that there does not exist any k-algebra having Gelfand-Kirillov dimension in the open interval (1,2).…”

Number theoretic techniques in the theory of Lie groups and differential geometry

“…Finally, we would like to mention the following theorem which provides a far-reaching generalization of the results of [2] and [11]. Theorem 10 ([18]).…”