On the existence of normal maximal subgroups in division rings

AbstractLet D be a division ring with centre F. Denote by D∗ the multiplicative group of D. The relation between valuations on D and maximal subgroups of D∗ is investigated. In the finite dimensional case, it is shown that F∗ has a maximal subgroup if Br(F) is non-trivial provided that the characteristic of F is zero. It is also proved that if F is a local or an algebraic number field, then D∗ contains a maximal subgroup that is normal in D∗. It should be observed that every maximal subgroup of D∗ contains either D′ or F∗, and normal maximal subgroups of D∗ contain D′, whereas maximal subgroups of D∗ do not necessarily contain F∗. It is then conjectured that the multiplicative group of any noncommutative division ring has a maximal subgroup

On subgroups in division rings of type 22

Let $D$ be a division ring with center $F$. We say that $D$ is a {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a ring.Comment: 10 pages, 0 figure

The AIQ Meta-Testbed: Pragmatically Bridging Academic AI Testing and Industrial Q Needs

AI solutions seem to appear in any and all application domains. As AI becomes more pervasive, the importance of quality assurance increases. Unfortunately, there is no consensus on what artificial intelligence means and interpretations range from simple statistical analysis to sentient humanoid robots. On top of that, quality is a notoriously hard concept to pinpoint. What does this mean for AI quality? In this paper, we share our working definition and a pragmatic approach to address the corresponding quality assurance with a focus on testing. Finally, we present our ongoing work on establishing the AIQ Meta-Testbed.Comment: Accepted for publication in the Proc. of the Software Quality Days 2021, Vienna, Austri

Tits alternative for maximal subgroups of GLn(D

Let D be a noncommutative division algebra of finite dimension over its centre F. Given a maximal subgroup M of GLn(D) with n ≥ 1, it is proved that either M contains a noncyclic free subgroup or there exists a finite family {Ki} r 1 of fields properly containing F with K ∗ i ⊂ M for all 1 ≤ i ≤ r such that M/A is finite if CharF = 0 and M/A is locally finite if CharF = p> 0, where A = K ∗ 1 × · · · × K ∗ r.

Free subgroups in maximal subgroups of GL1(D

Let D be a division algebra of finite dimension over its centre F. Given a noncommutative maximal subgroup M of D ∗: = GL1(D), it is proved that either M contains a noncyclic free subgroup or there exists a maximal subfield K of D which is Galois over F such that K ∗ is normal in M and M/K ∗ ∼ = Gal(K/F). Using this result, it is shown in particular that if D is a noncrossed product division algebra, then M does not satisfy any group identity.