On the radicality of maximal subgroups in

Low Diameter Algebraic Graphs

“…In the sequel, we compare Theorem 1 with known results when D = F is a field [2]. A division ring D with center F is weakly locally finite if for every finite subset S of D the division subring F (S) generated by S over F is a finite dimensional vector space over its center [9]. Theorem 2 below generalizes [2, Theorem 4.2], which holds for n ≥ 2 and D = F a field.…”

Subnormal subgroups of almost locally simple artinian algebras with involutions

“…The problem of finding a suitable algebraic division ring which is not locally finite remains unsolved until now and it is often referred to as the Kurosh Problem for division rings [18,Problem K]. Recall that a division ring D is said to be weakly locally finite if for every finite subset S of D, the division subring P(S) of D generated by S over the prime subfield P of D is centrally finite (see [9]). The following lemma provides an important property of weakly locally finite division rings.…”

Free subgroups in almost subnormal subgroups of general skew linear groups

“…The aim of this section is to show that if a non-commutative division ring D is weakly locally finite, then every non-central almost subnormal subgroup of D * contains a non-cyclic free subgroup. Recall that a division ring D is weakly locally finite if every finite subset in D generates a centrally finite division subring in D. Some basic properties and the existence of non-cyclic free subgroups in weakly locally finite division rings can be seen in [8] and [16]. The following lemma is useful for our next study.…”