On normal subgroups of division rings which are radical over a proper division subring

Bien, Mai Hoang; Duong, Dung Hoang

doi:10.1556/sscmath.51.2014.2.1265

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…The implications (4) ⇒ (3) ⇒ (2) ⇒ (1) are trivial. Now we will show that (1) implies (4). Assume that N is a non-central almost subnormal subgroup of G and…”

Section: Theorem a Let D Be A Division Ring That Is Not A Locally Fimentioning

confidence: 96%

Free subgroups in almost subnormal subgroups of general skew linear groups

Ngoc

Bien

Hai

2017

St. Petersburg Math. J.

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Abstract. Let D be a weakly locally finite division ring and n a positive integer. In this paper, we investigate the problem on the existence of non-cyclic free subgroups in non-central almost subnormal subgroups of the general linear group GLn(D). Further, some applications of this fact are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of GLn(D) are described.

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“…The implications (4) ⇒ (3) ⇒ (2) ⇒ (1) are trivial. Now we will show that (1) implies (4). Assume that N is a non-central almost subnormal subgroup of G and…”

Section: Theorem a Let D Be A Division Ring That Is Not A Locally Fimentioning

confidence: 96%

Free subgroups in almost subnormal subgroups of general skew linear groups

Ngoc

Bien

Hai

2017

St. Petersburg Math. J.

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“…However, in the case when m = 1, that is, GL 1 (D) = D * , there are division rings D whose multiplicative groups D * contain non-normal subnormal subgroups [15]. If Conjecture 1 holds, then one would extend trivially several results on normal subgroups for subnormal subgroups in division rings (e. g., results in [7,12,13,18]). The Conjecture 1 is affirmative in case D is finite dimensional over a p-local field with p = 2 [14].…”

Section: Introductionmentioning

confidence: 99%

“…The most important result concerning the structure of subnormal subgroups was obtained by Stuth in 1964 asserting that (i) If G is a noncentral subnormal subgroup of D * and x G is the conjugacy class of the noncentral element x ∈ D * in G, then the division subring generated by x G is D, (ii) Every soluble subnormal subgroup of D * is central. In recent years, for a division ring D, there has been renewed interest in the study of normal subgroups of D * [7,12,13,18]. Herstein and Scott conjectured that every subnormal subgroup of D * is normal in D * (see [17,Page 80]) and it was shown in [15] that this conjecture of Herstein and Scott holds for the real quaternion division ring, but not for a finite dimensional division ring D over a p-local field.…”

Section: Introductionmentioning

confidence: 99%

“…Whether can one conclude the algebraicity of a division ring over its center from the algebraicity of its maximal subfields over the center? The second named author and D.H. Dung in [7], could provide a nonalgebraic division ring which contains a maximal subfield algebraic over the center and answer the question in negative. In fact, using the Mal'cev-Neumann's construction of Laurent series rings, they present an example of a division ring D whose all elements are Kurosh, that is for each element x ∈ D there exists a centrally finite division subring of D containing x as a non-central element, but D is not algebraic.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Self-Invariant Maximal Subfields and Their Connexion with Some Conjectures in Division Rings

Aaghabali¹,

Bien²

2019

Preprint

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Let D be a division algebra with center F . A maximal subfield of D is defined to be a fieldThis kind of subfields is important because they have strong connexion with most famous Albert's Conjecture (every division ring of prime index is cyclic). In fact, we pose a question that asserts whether every division ring whose all maximal subfields are self-invariant has to be commutative. The positive answer to this question, in finite dimensional case, implies the Albert's Conjecture (see §2). Although we show the Mal'cev-Neumann division ring demonstrates negative answer in the case of infinite dimensional division rings, but it is still most likely the question receives positive answer if we restrict ourselves to the finite dimensional division rings. We also have had the opportunity to use the Mal'cev-Neumann structure to answer Conjecture 1 below in negative (see §3). Finally, among other things, we rely on this kind of subfields to present a criteria for a division ring to have finite dimensional subdivision ring (see §4).

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