Free groups in a normal subgroup of the field of fractions of a skew polynomial ring

Gonçalves, Jairo Z.

doi:10.1080/00927872.2017.1298774

Cited by 4 publications

(2 citation statements)

References 12 publications

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“…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%

“…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%

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Aaghabali¹,

Bien²

2019

Preprint

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Subnormal subgroups and self-invariant maximal subfields in division rings

Aaghabali

Bien

2021

Journal of Algebra

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Free symmetric pairs in the field of fractions of enveloping Lie algebras with involution

Goncalves

2022

J. Algebra Appl.

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Let [Formula: see text] be a division ring with center [Formula: see text], [Formula: see text], let [Formula: see text] be an involution of [Formula: see text], and let [Formula: see text] be the multiplicative group of [Formula: see text]. A pair [Formula: see text] is called free symmetric, if it is formed by symmetric elements, and it generates a free non-cyclic subgroup of [Formula: see text]. If [Formula: see text] is the enveloping algebra of the non-abelian nilpotent Lie [Formula: see text]-algebra [Formula: see text] over the field [Formula: see text] of characteristic [Formula: see text], and [Formula: see text] is a [Formula: see text]-involution of [Formula: see text] extended to the field of fractions [Formula: see text] of [Formula: see text], we show that [Formula: see text] contains free symmetric pairs. We also discuss the consequences of symmetric elements of a normal subgroup being torsion over the center.

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Free groups in a normal subgroup of the field of fractions of a skew polynomial ring

Cited by 4 publications

References 12 publications

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

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

Subnormal subgroups and self-invariant maximal subfields in division rings

Free symmetric pairs in the field of fractions of enveloping Lie algebras with involution

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