Article
Kyungpook Mathematical Journal 2023; 63(2): 187198
Published online June 30, 2023 https://doi.org/10.5666/KMJ.2023.63.2.187
Copyright © Kyungpook Mathematical Journal.
Quasinormal Subgroups in Division Rings Radical over Proper Division Subrings
Le Qui Danh∗, Trinh Thanh Deo
Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam
Vietnam National University, Ho Chi Minh City, Vietnam
University of Architecture Ho Chi Minh City, 196 Pasteur Str., Dist. 3, Ho Chi Minh City, Vietnam
email : danh.lequi@uah.edu.vn
Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam
Vietnam National University, Ho Chi Minh City, Vietnam
email : ttdeo@hcmus.edu.vn
Received: August 5, 2022; Revised: February 12, 2023; Accepted: May 4, 2023
Abstract
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
The motivation for this study comes from a question posed by I.N. Herstein in the Israel Journal of Mathematics in 1978. Specifically, let
Keywords: division ring, normal subgroup, quasinormal subgroup, Mal’cevNeumann, radical
1. Introduction
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
The motivation of this study comes from a question posed by Herstein [13] in 1978: Is it true that every subnormal subgroup of the multiplicative group of a division ring

• for periodic subgroups [13, Theorem 8];

• for a division ring with uncountable center [14, Theorem 2];

• for a division ring finite dimensional over the center [9, Theorem 1];

• for normal subgroups in a division ring of type 2, where the center
F is replaced by an arbitrary proper division subringK ofD [8, Theorem 3.2].
Motivated by the results obtained in [8][Theorem 3.2], the authors posed a more general question: For a division ring

(1)
D is weakly locally finite; 
(2)
F is uncountable; 
(3)
D is the Mal'cevNeumann division ring.
Recall that, in the theory of division rings, there are several classical constructions of new division rings from given ones. One of such structures is the class of Mal'cevNeumann division rings, were completely presented in [18] by Neumann who used Mal'cev's ideas in [17]. Mal'cevNeumann division rings have a vast number of applications. For example, they were used to construct examples of noncrossed product division algebras [2, 10, 11], to describe the multiplicative group of group rings of ordered groups, [15, Corollary 14.24], etc. The problems describe the properties of Mal'cev Neumann division rings and their special cases have been studied in several papers. For instance, it was proved that there are free group algebras in the Mal'cevNeumann division rings [19]; there are free symmetric group algebras in division rings generated by polyorderable groups [6]. Also, Amitsur and Tignol determined abelian Galois subfields of the Mal'cevNeumann division rings with
The paper is organized as follows: In Section 2, we prove that
The symbols and notations we use in this paper are standard that can be found in the literature on division rings and in the cited items of the presented paper.
2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
Recall that a division ring
In [8, Theorem 3.2], it was shown that every normal subgroup of the multiplicative group of a division ring of type 2 which is radical over its proper division subring is central. The proof of [8, Theorem 3.2] is based on the properties of a division subring generated by two elements over
Theorem 2.1. Let
Clearly, we can assume
With
There are two possible subcases for a given element
Then,
This implies
Hence,
Since
Since
Take
Thus, the claim is proved.
Therefore, for each
Using Theorem 2.1, we can now extend some earlier results. In [14], Herstein proved that in a division ring
Theorem 2.2. Let
3. Herstein Conjecture for Quasinormal Subgroups
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
Based on the definition of the weakly locally finite division rings in [4], in this section, we will show a similar result to a conjecture of Herstein for quasinormal subgroups. First, we use the following result on the relationship between quasinormal subgroups, radical subgroups, and subnormal subgroups.
Recall that a subgroup
A
Lemma 3.1. Let
Next, combining [4, Theorem 11] and Theorem 2.2, we have the following result for the normal subgroup of the multiplicative group of a division ring.
Theorem 3.2. Let
For a group
To prove the main result of this section, we need the following results from [7, Theorem 1], [21, Theorem 4], and [5, Theorem B].
Lemma 3.3. Let
Lemma 3.4.([21, Theorem 4]) Let
The following is an interesting result of Faith in [5] regarding a division ring radical over its proper division subring.
Lemma 3.5.([5, Theorem B]) Let
Now we show the main result of this section.
Theorem 3.6. Let
In both cases,
4. Some More Results on Quasinormal Subgroups
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
In this section, we will give some results regarding quasinormal subgroups of
Lemma 4.1. Let
Let
Lemma 4.2. Let
Lemma 4.3. Let
Thus,
Hence,
Lemma 4.4. Let
Using Lemma 4.3, by the similar way, we can get
Now we can prove an analogue of [16, Corollary 3] for a quasinormal subgroup which is radical over a proper division subring.
Theorem 4.5. Let
Theorem 4.6. Let
2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
Recall that a division ring
In [8, Theorem 3.2], it was shown that every normal subgroup of the multiplicative group of a division ring of type 2 which is radical over its proper division subring is central. The proof of [8, Theorem 3.2] is based on the properties of a division subring generated by two elements over
Theorem 2.1. Let
Clearly, we can assume
With
There are two possible subcases for a given element
Then,
This implies
Hence,
Since
Since
Take
Thus, the claim is proved.
Therefore, for each
Using Theorem 2.1, we can now extend some earlier results. In [14], Herstein proved that in a division ring
Theorem 2.2. Let
Acknowledgments.
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
The authors would like to thank the referee for his/her useful suggestions. This work was written while the second author was working at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The author would like to thank the institute for providing a fruitful research environment and working condition.
Footnote
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
This research is funded by University of Architecture Ho Chi Minh City under grant number KHCB 03  NCKH 22.
References
 Abstract
 1. Introduction
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 3. Herstein Conjecture for Quasinormal Subgroups
 4. Some More Results on Quasinormal Subgroups
 2. Normal Subgroups in Division Rings that are Radical over Proper Division Subrings
 Acknowledgments.
 Footnote
 References
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