Locally solvable subnormal and quasinormal subgroups in division rings

Abstract

In this paper, we show that every locally solvable subnormal subgroup or locally solvable quasinormal subgroup of the multiplicative group of a division ring is central.

Intersection graphs of quasinormal subgroups of general skew linear groups

“…Lemma 5.2 fully describes the case Diam(Γ q (GL n (D))) = 0, and so we have the assertions (4), ( 5), and (6). Hence, by Theorem 4.8 and 4.9, we have the assertions ( 7), ( 8), (9), and (10).…”

“…In [8] and [9], authors proved some properties of quasinormal subgroups of GL n (D). We have the following results.…”

“…The main object of this paper is Γ q (G), the subgraph of Γ(G) induced on the nontrivial proper quasinormal subgroups of G. Recall that a subgroup Q of G is quasinormal (also permutable) in G if QH = HQ for every H ≤ G. Many interesting properties of quasinormal subgroups have already been discovered, for example, [12,18]. Recently, quasinormal subgroups of the general linear group GL n (D) over a division ring is researched in [8,9]. Continuing this topic, we study on Γ q (G) in the generality and in the case G = GL n (D), the general linear group over a division ring D.…”

“…We can suppose that D is noncommutative. By[9, Lemma 8], G is nonabelian and is an absolutely irreducible subgroup in D * , that isD = F [G]. Suppose that N is a locally soluble normal subgroup of G. By [20, 4.Corollary], N contains an abelian normal subgroup A of G such that N/A locally finite.…”

A Note on Subnormal Subgroups in Division Rings Containing Solvable Subgroups

“…After that, in [8], Conjecture 1.3 holds if D is a weakly locally finite division ring, that is, D satisfies the condition: for every finite subset S of D , the division subring generated by S and a prime subfield P of D is finite-dimensional over the centre . Recently, Conjecture 1.3 was fully confirmed in [3, Theorem 1]. By applying the Main Theorem, we can also show that Conjecture 1.3 is true.…”