A note on the existence of non-cyclic free subgroups in division rings

Free subgroups in almost subnormal subgroups of general skew linear groups

“…Later, Chiba [7] proved the same result but without the assumption on the existence of such an element a in D. In [11], Gonçalves proved that any non-central subnormal subgroup of D * contains a non-cyclic free subgroup provided D is centrally finite. Recently, B. X. Hai and N. K. Ngoc [16] proved the same result for weakly locally finite division rings. Recall that a division ring D is called weakly locally finite if every finite subset of D generates a centrally finite division subring.…”

“…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.…”

“…The class of weakly locally finite division rings we consider in Theorem 4.5 is very large. Indeed, in [16], it was indicated that this class strictly contains the class of locally finite division rings. Recently, in [17], we have constructed infinitely many examples of weakly locally finite division rings that are not even algebraic over the center.…”

“…Recall that a division ring D is called weakly locally finite if every finite subset of D generates a centrally finite division subring. It was proved that every locally finite division ring is weakly locally finite, and there exist infinitely many weakly locally finite division rings that are not even algebraic over their centers (see [16] and [17]), so they are not locally finite. Logically, it is natural to carry over the results above for subnormal subgroups of GL 1 (D) to that of GL n (D), n ≥ 2 (see [14,24,30]).…”

On division subrings normalized by almost subnormal subgroups in division rings

“…Since α(h, gw r (x, e))y − yα(h, gw r (x, e)) is non-zero in D(x, y), D * satisfies generalized rational identity α(h, gw r (x, e))y − yα(h, gw r (x, e)) = 0. Therefore, by Lemma 3.1, D is centrally finite, so in view of [5,Theorem 3], K is centrally finite. By Case 1, K = D. But this fact contradicts the assumption that N r ⊆ K. Thus, N r ⊆ K, and the claim is proved.…”