Locally solvable maximal subgroups in division rings

Bui Xuan Hai; Huynh Viet Khanh

REVIEW 4 major objections 4 minor 32 references

Locally solvable maximal subgroups in division rings

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper establishes that every locally solvable almost subnormal subgroup of a division ring's multiplicative group lies in the center, and that a non-abelian locally solvable maximal subgroup forces the ring to be a cyclic algebra of…

desk verdict Genuinely new results on locally solvable almost subnormal subgroups in division rings, but the main classification has a load-bearing gap in the finite-dimensionality reduction. read the letter →

arxiv 1908.04925 v3 pith:GGR7UIIW submitted 2019-08-14 math.RA math.GR

Huynh Viet Khanh , Bui Xuan Hai This is my paper · ORCID

classification math.RAmath.GR MSC 16K2020F19

keywords divisionringslocallysolvablesubgroupsalmostsubnormalmaximalcyclicalgebrascentralcrossedproductsskewlineargroups

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves a structural dichotomy for almost subnormal subgroups of the multiplicative group of a division ring. On one side, every locally solvable almost subnormal subgroup lies in the center of the ring, extending known central-subgroup theorems from subnormal to almost subnormal chains. On the other side, if an almost subnormal subgroup contains a non-abelian locally solvable maximal subgroup, then the whole division ring is a cyclic algebra of prime degree p: there is a maximal subfield K of degree p over the center F, with [D:F] = $p^{2}$ and D generated by K and an element x with x^p in F. The paper also proves that locally nilpotent maximal subgroups are always abelian. These results matter because they show that a locally solvable maximal subgroup can be non-abelian only in the smallest possible non-commutative cyclic-algebra setting.

What carries the argument

The argument rests on the notion of an almost subnormal subgroup: a subgroup H of D* connected to D* by a finite chain of subgroups in which each step is either a normal inclusion or a finite-index inclusion. The main technical engine is a collection of crossed-product and normalizer criteria for skew linear groups, which let the authors move from a normal subgroup A of the maximal subgroup M to the division ring generated by A, showing that either A is abelian or it generates all of D (Lemma 3.6). Finite dimensionality is established through an omitted modification of a known theorem on metabelian maximal subgroups (Proposition 3.7) and through case analysis on the periodic radical, after which the finite-dimensional case is handled by viewing M as a subgroup of GL_n(F) and applying the solvable maximal subgroup classification of Theorem 3.9. The decisive structural step is the reduction of the locally solvable case to the solvable case, where Galois theory produces the cyclic Galois extension K/F and the direct-sum decomposition D = ⊕ Kx^i.

What would settle it

Find a division ring D with center F and an almost subnormal subgroup G of D* containing a non-abelian metabelian maximal subgroup M with [D:F] infinite. If such an example exists, the finite-dimensionality step used in Theorems 3.9 and 3.11 collapses; checking whether the omitted modification of [9, Theorem 3.3] remains valid when the almost subnormal chain includes a finite-index step would settle the matter.

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Extended reading notes

Core claim

The central claim is Theorem 3.11: if G is an almost subnormal subgroup of D* and M is a non-abelian locally solvable maximal subgroup of G, then there is a maximal subfield K of D such that K/F is finite Galois, Gal(K/F) is isomorphic to M/(K* ∩ G) and to Z_p for some prime p, [D:F] = $p^{2}$, D = F[M] = K ⊕ Kx ⊕ ... ⊕ $Kx^{{p-1}}$, and x^p ∈ F for any x ∈ M \ K. In the same setting, Theorem 2.15 states that any locally solvable almost subnormal subgroup of D* is contained in F, and Theorem 3.10 states that a locally nilpotent maximal subgroup is abelian. Together these results characterize when the classical central-subgroup theorem for solvable subnormal subgroups survives in the almost subnormal setting: locally solvable almost subnormal subgroups are central, and the only way a maximal subgroup can be non-abelian and locally solvable is through a cyclic algebra of prime degree.

Load-bearing premise

The load-bearing premise is the unproved Proposition 3.7, which asserts that a non-abelian metabelian maximal subgroup of an almost subnormal subgroup makes the division ring finite-dimensional over its center; the proof is merely said to follow by a simple modification of a known theorem.

Editorial extensions

If this is right

In any division ring, every locally solvable almost subnormal subgroup of the multiplicative group is central, so no non-central subgroup of this kind can exist.
A non-abelian locally solvable maximal subgroup can occur only when the division ring is a cyclic algebra of prime degree p with [D:F] = p^2.
In the maximal subgroup M, the subgroup K* ∩ G is both the FC-center and the Hirsch-Plotkin radical, and the quotient M/(K* ∩ G) is cyclic of prime order p.
Every element x of M outside K satisfies x^p ∈ F, and the division ring is generated by M together with F, so the entire ring is controlled by the maximal subgroup's structure.
Locally nilpotent maximal subgroups of almost subnormal subgroups are always abelian, which rules out a broad class of potential non-abelian examples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the proof's finite-dimensionality step relies wholly on Proposition 3.7, whose proof is omitted; if that omitted modification fails for almost subnormal chains with finite-index steps, the classification of Theorems 3.9 and 3.11 loses its foundation.
Editorial extension: the theorem suggests a general dichotomy for almost subnormal subgroups of division rings—either the subgroup is central, or it generates the entire division ring and forces a cyclic algebra structure—so one might test whether similar dichotomies hold for other group classes such as locally finite or locally graded subgroups.
Editorial extension: because the quotient M/(K* ∩ G) is cyclic of prime order, the maximal subgroup is built from a single cyclic extension step; a natural next question is whether solvable-by-finite maximal subgroups of almost subnormal subgroups admit an analogous description.
Editorial extension: the removal of the algebraicity hypothesis on M' (which earlier results required) is a direct consequence of the finite-dimensionality step, and a reader could try to construct a skew Laurent division ring example to test whether the finite-dimensional conclusion really needs maximality or merely local solvability.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper makes a real advance: it proves the almost-subnormal version of the locally solvable central subgroup theorem (Theorem 2.15) and extends the quaternionic-style classification of non-abelian solvable maximal subgroups to locally solvable ones (Theorem 3.11). That is the right next step after [2], [9], [19], and the main theorems are genuinely new. The paper is also honest about using heavy imported machinery (Wehrfritz crossed-product criteria, Stuth, Robinson) and is generally careful with normalizer arguments. The reader's report is fair: the chain of reasoning is detailed and plausible.

But there is a real hole, and it is load-bearing. In Theorem 3.10 Case 2, after proving M/Z is simple, the proof says 'Because M/Z is locally nilpotent, we conclude that M/Z is finite of prime order ([24, 12.5.2, p.367])'. That inference is not valid. There are infinite simple locally finite p-groups (McLain groups), and every locally finite p-group is locally nilpotent. Robinson 12.5.2 cannot say what they need. The same step appears in Theorem 3.11 for M/B, where M/B is not even shown locally nilpotent. Both theorems use this step to get [D:F] finite and then reduce to GL_n(F). Without a division-ring-specific argument, the finite-dimensionality reduction is unsupported. This is not a minor typo; it is the hinge of the maximal-subgroup classification.

A secondary soft spot is Proposition 3.7: the proof is omitted as a 'simple modification' of [9, Theorem 3.3], and it is exactly what supplies finite dimensionality in Theorem 3.9. A referee should ask for the modification to be written out. Also the final step 'view M as subgroup of GL_n(F) to conclude it is solvable' is a known theorem but cited nowhere; that is minor.

Is the damage fatal? Not necessarily. The main result is likely true, and the gap may be patchable by a stronger division-ring argument or a correction of the simple-quotient step. But as it stands, Theorems 3.10 and 3.11 are not proved. This paper is for specialists in division rings, skew linear groups, and subgroup structure of D*. They will want to see the gap fixed. It deserves a serious referee, but it is not ready to be accepted as is.

Referee Report

4 major / 4 minor

Summary. The paper studies the multiplicative group D* of a division ring D with center F. Theorem 2.15 asserts that every locally solvable almost subnormal subgroup of D* is contained in F. The main results concern maximal subgroups M of an almost subnormal subgroup G: Theorem 3.10 asserts that if M is locally nilpotent then M is abelian, and Theorem 3.11 asserts that if M is non-abelian and locally solvable then D is a cyclic algebra of prime degree p^2 over F with Gal(K/F) ≅ M/K*∩G ≅ Z_p, and D = F[M] = ⊕ Kx^i with x^p ∈ F for x ∈ M\K. The proof strategy is to establish finiteness of [D:F], view M as a linear group over F, apply a solvable-linear-group result, and then invoke Theorem 3.9.

Significance. If valid, Theorem 2.15 resolves a natural generalization of Stuth's theorem on solvable subnormal subgroups, and Theorems 3.10 and 3.11 give a sharp structural classification of locally solvable (and locally nilpotent) maximal subgroups, extending earlier work on solvable maximal subgroups in division rings. The paper is clearly written and uses established techniques from skew linear groups. However, the classification theorems currently depend on two unsupported steps: an omitted proof of Proposition 3.7 and an invalid group-theoretic inference used to obtain finiteness of [D:F]. These are load-bearing, so the main results are not yet established as written.

major comments (4)

[Section 3, Proposition 3.7] The proof of Proposition 3.7 is omitted, with only the statement that it is a 'simple modification' of [9, Theorem 3.3]. This proposition is used directly in Theorem 3.9, Case 3, to conclude [D:F] < ∞, and it is inherited by Theorem 3.11 through the reduction to solvable linear groups. A central lemma on which the main finite-dimensionality step rests cannot be left unproved; the adaptation to the almost subnormal setting must be written out in full.
[Theorem 3.10, Case 2] The proof asserts: 'Because M/Z is locally nilpotent, we conclude that M/Z is finite of prime order ([24, 12.5.2, p.367]).' This inference is false as a general group-theoretic statement. For example, McLain groups are infinite simple locally finite p-groups, hence locally nilpotent, but they are not cyclic of prime order. The proof invokes no division-ring-specific property at this point, so the step leading to [D:F] < ∞ is unsupported.
[Theorem 3.11, Case 2] The proof repeats the same invalid inference for M/B: 'Again by [24, 12.5.2, p.367], we have M/B is a finite group of prime order.' Moreover, M/B is a quotient of a locally solvable group by its Hirsch-Plotkin radical; nothing in the argument shows that M/B is locally nilpotent, and in general such a quotient need not be locally nilpotent at all. Thus the finite-dimensionality reduction in the main theorem is not valid without a new argument that uses the division-ring structure.
[Theorem 3.9, Case 3] This case concludes [D:F] < ∞ from Proposition 3.7, so the omitted proof of Proposition 3.7 is load-bearing for Theorem 3.9. Since Theorem 3.11 reduces to Theorem 3.9 after proving finite dimensionality, the main classification inherits the gap. The authors must either supply a complete proof of Proposition 3.7 or replace this route with a directly verified finite-dimensionality argument.

minor comments (4)

[Theorem 3.11, Case 2] In the sentence 'let C be a normal subgroup of M properly containing C', the second 'C' should be 'B'.
[Theorem 3.11, Case 2] The phrase 'B/T is the Hirsch-Plotkin radical of the group G/T' should read 'of the group M/T'.
[Theorem 3.10, proof] The displayed isomorphism 'D ⊗F Dop ≅ Mn(D)' appears to be a typo; for a central division algebra of degree n, the standard isomorphism is D ⊗_F D^{op} ≅ End_F(D) ≅ M_{n^2}(F), and the subsequent embedding into GL_n(F) should be clarified as the regular representation into GL_{n^2}(F), which still makes M a linear group.
[Throughout] There are several typographical errors, including 'divsision' in Lemma 2.6 and 'Asumme' in Lemma 2.8, which should be corrected.

Circularity Check

1 steps flagged · score 4.0 of 10

Finite-dimensionality premise in Theorem 3.9 is imported from an omitted 'simple modification' of the authors' earlier theorem; otherwise the derivation is not circular.

self citation load bearing [Section 3, Proposition 3.7; invoked in Theorem 3.9 (Case 3) and inherited by Theorem 3.11]
"The proof of the following proposition is a simple modiﬁcation of the pro of of [9, Theorem 3.3], so it should be omitted. Proposition 3.7. Let D be a division ring with center F , and G an almost subnormal subgroup of D∗ . If M is a non-abelian metabelian maximal subgroup of G, then [D : F ] < ∞."

Proposition 3.7 is the only step that supplies [D:F] < ∞ in the metabelian case, and its proof is not given: it is replaced by a 'simple modification' of [9, Theorem 3.3], a paper coauthored by B. X. Hai, one of the present authors. In Theorem 3.9, Case 3, the finite-dimensionality conclusion follows directly from Proposition 3.7, and Theorem 3.11 inherits that step when it reduces the locally solvable case to Theorem 3.9. Since the modification is not shown and the cited theorem is the authors' own prior work, the central classification rests on a load-bearing self-citation rather than on a derivation contained in this paper.

full rationale

The main results are not circular in the sense of defining the target conclusion into the hypotheses. Theorem 2.15 is built from Cartan-Brauer-Hua-type results and Wehrfritz's crossed-product lemmas, and the classification in Theorems 3.9-3.11 is a case analysis that invokes the authors' earlier [10, Theorem 3.1] only after [D:F] < ∞ has been established. The principal circularity-sensitive point is Proposition 3.7, whose omitted proof is deferred to a same-author prior theorem; this is load-bearing because finite dimensionality is the key reduction to the skew-linear setting. I therefore set the score at 4 rather than 0: there is some self-citation at a load-bearing step, but the central claim retains substantial independent content. The apparent group-theoretic gap in Theorem 3.10/3.11, where a simple locally nilpotent quotient is asserted to be finite of prime order via [24, 12.5.2, p.367], is a correctness concern (infinite simple locally finite p-groups exist), not a circularity, since no quantity is defined in terms of the conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted numerical parameters and no newly invented entities appear. The central claims rest on a large stock of imported theorems, none of which is proved in this paper. The only explicit local derivation gap is Proposition 3.7, whose proof is omitted, and the unstated local-solvable linear group theorem is an additional uncited assumption.

assumptions (6)

standard math Wehrfritz normalizer theorem [33, 1.4], quoted as Lemma 2.1
Imported without proof; used in Lemma 2.2 to show H/GF* is solvable for locally nilpotent G.
standard math Stuth's generalization of Cartan-Brauer-Hua [26, Theorem 1]
Used in Theorem 2.3 to conclude that a division subring normalized by a non-central subnormal subgroup is either F or D.
standard math Ore domain and crossed product criteria [30, Cor 24], [29, Point 20], [27, 2.5], [28, 7], [31, 3.2], and [31, Theorem 1.1(c)]
Used repeatedly in Lemmas 2.12-2.14 and Theorems 3.9-3.11 to identify normalizers, Ore domains, and crossed product structures.
standard math Wehrfritz normalizer classification [32, Prop 4.1], quoted as Lemma 3.5
Used in Theorem 3.9 to split the normalizer computation into three cases based on the structure of the periodic normal subgroup.
standard math Free subgroup and group identity results [22, Theorem 2.2], [10, Theorem 3.1], and [16, Theorem 8]
Imported to force centrality or finite dimensionality in the almost subnormal setting.
domain assumption Locally solvable subgroups of GL_n(F) are solvable
Invoked in the final paragraph of Theorem 3.11 to conclude M is solvable after viewing it in GL_n(F); no citation is provided for this nontrivial linear-group fact.

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Pith. "Pith review of Locally solvable maximal subgroups in division rings." pith.science (2026). https://pith.science/paper/GGR7UIIW

@misc{pith2026190804925,
  author       = {Pith},
  title        = {Pith review of: Locally solvable maximal subgroups in division rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GGR7UIIW}},
  note         = {Machine review of arXiv:1908.04925}
}

abstract

Let $D$ be a division ring with center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ contains a non-abelian locally solvable maximal subgroup, then $D$ must be a cyclic algebra of prime degree over $F$. Moreover, it is proved that every locally nilpotent maximal subgroup of $G$ is abelian.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

32 extracted references · 32 canonical work pages

[9]

B. X. Hai and N. A. Tu. On multiplicative subgroups in divi sion rings, J. Algebra Appl. 15(3) (2016), 1650050 (16 pages)

work page 2016
[19]

H. V. Khanh, On locally solvable subgroups in division r ings, arXiv:1903.11216v3 [math.RA]

work page Pith review arXiv 1903
[1]

Akbari, R

S. Akbari, R. Ebrahimian, H. Momenaei Kermani, A. Salehi Golseﬁdy, Maximal subgroups of GLn(D), J. Algebra 259 (2003) 201-225

work page 2003
[3]

T. T. Deo, M. H. Bien, B. X. Hai, On division subrings norma lized by almost subnormal subgroups in division rings, Period. Math. Hung. (2019); DO I: 10.1007/s10998-019-00282-5

work page doi:10.1007/s10998-019-00282-5 2019
[4]

T. H. Dung, A note on locally soluble almost subnormal sub groups in divsion rings, Int. J. Group Theory (2019), DOI: 10.22108/IJGT.2019.116399.154 6

work page arXiv 2019
[5]

H. R. Dorbidi, R. Fallah-Moghaddam, M. Mahdavi-Hezaveh i, Soluble maximal subgroups in GLn(D), J. Algebra Appl. (6)10 (2011) 1371-1382

work page 2011
[6]

Ebrahimian, Nilpotent maximal subgroups of GLn(D), J

R. Ebrahimian, Nilpotent maximal subgroups of GLn(D), J . Algebra 280 (2004) 244248

work page 2004
[7]

Fallah-Moghaddam, Maximal subgroups of SL n(D), J

R. Fallah-Moghaddam, Maximal subgroups of SL n(D), J. Algebra 531 (2019) 70-82

work page 2019

Show all 32 references

[8]

B. X. Hai, H. V. Khanh, and M. H. Bien, Generalized power ce ntral group identities in almost subnormal subgroups of Algebra i Analiz 34:1 (2019) 225-239 (Russian), English transl. in St. Petersburg Math. J. (2019) (accepted)

2019
[10]

B. X. Hai and H. V. Khanh, Free subgroups in maximal subgr oups of skew linear groups, Internat. J. Algebra Comput. 29(3) (2019) 603-614

2019
[11]

B. X. Hai, D. V. P. Ha, On locally soluble maximal subgrou ps of the multiplicative group of a division ring, Vietnam J. Math. 38(2) (2010) 237- 247

2010
[12]

B. X. Hai and N. V. Thin, On subnormal subgroups in genera l skew linear groups, Vestnik St. Petersburg University. Mathematics, 46 (2013) 43-48

2013
[13]

B. X. Hai and N. V. Thin, On locally nilpotent subgroups o f GL 1(D), Comm. Algebra 37 (2009) 712-718

2009
[14]

Hartley, Free groups in normal subgroups of unit grou ps and arithmetic groups, Contemp

B. Hartley, Free groups in normal subgroups of unit grou ps and arithmetic groups, Contemp. Math. 93 (1989) 173-177

1989
[15]

Hazrat, M

R. Hazrat, M. Mahdavi-Hezavehi and M. Motiee, Multipli cative groups of division rings, Math. Proc. R. Ir. Acad. 114A (2014) 37-114

2014
[16]

I. N. Herstein. Multiplicative commutators in divisio n rings. Israel J. Math. 31 (1978) 180-188

1978
[17]

L. K. Hua, On the multiplicative group of a ﬁeld, Acad. Si nica Science Record (1950)

1950
[18]

M. S. Huzurbazar, The multiplicative group of a divisio n ring, Soviet Math. Dokl. (1960) 1433-1435

1960
[20]

H. V. Khanh, B. X. Hai, A note on solvable maximal subgrou ps in subnormal subgroups of GLn(D), arXiv:1809.00356v2 [math.RA]

arXiv
[21]

T. Y. Lam, A First Course in Noncommutative Rings, 2nd ed n, GTM 131, Springer-Verlag, New York, 2001

2001
[22]

N. K. Ngoc, M. H. Bien, B. X. Hai, Free subgroups in almost subnormal subgroups of general skew linear groups, Algebra i Analiz 28(5) (2016) 220-235, t ranslation in St. Petersburg Math. J. 28(5) (2017) 707-717

2016
[23]

Ramezan-Nassab and D

M. Ramezan-Nassab and D. Kiani, Nilpotent and polycycl ic-by-ﬁnite maximal subgroups of skew linear groups, J. Algebra 399 (2014) 269-276

2014
[24]

D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed n, Springer, 1995

1995
[25]

Shirvani and B

M. Shirvani and B. A. F. W ehrfritz, Skew Linear Groups, C ambridge Univ. Press, 1986

1986
[26]

C. J. Stuth, A generalization of the Cartan-Brauer-Hua Theorem, Proc. Amer. Math. Soc. 15(2) (1964) 211-217

1964
[27]

B. A. F. W ehrfritz, Locally nilpotent skew linear group s, Proc. Edinburgh Math. Soc. 29 (1986) 101-113

1986
[28]

B. A. F. W ehrfritz, Locally nilpotent skew linear group II, Proc. Edinburgh, Math. So. 30(3) (1987) 423-426

1987
[29]

B. A. F. W ehrfritz, Soluble and locally soluble skew lin ear groups, Arch. Math. 49 (1987) 379-388

1987
[30]

B. A. F. W ehrfritz, Goldie subrings of Artinian rings ge nerated by groups, Q. J. Math. Oxford 40 (1989) 501-512. ALMOST SUBNORMAL SUBGROUPS 13

1989
[31]

B. A. F. W ehrfritz, Crossed product criteria and skew li near groups, J. Algebra 141 (1991) 321353

1991
[32]

B. A. W ehrfritz, Normalizers of nilpotent subgroups of division rings, Q. J. Math. 58 (2007) 127-135

2007
[33]

B. A. F. W ehrfritz, Normalizers of subgroups of divisio n rings, J. Group Theory 11 (2008) 399413. E-mail address : huynhvietkhanh@gmail.com; bxhai@hcmus.edu.vn F aculty of Mathematics and Computer Science, VNUHCM - University of Science, 227 Nguyen V an Cu Str., Dist. 5, Ho ...

2008

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [9]

B. X. Hai and N. A. Tu. On multiplicative subgroups in divi sion rings, J. Algebra Appl. 15(3) (2016), 1650050 (16 pages)

work page 2016

[2] [19]

H. V. Khanh, On locally solvable subgroups in division r ings, arXiv:1903.11216v3 [math.RA]

work page Pith review arXiv 1903

[3] [1]

Akbari, R

S. Akbari, R. Ebrahimian, H. Momenaei Kermani, A. Salehi Golseﬁdy, Maximal subgroups of GLn(D), J. Algebra 259 (2003) 201-225

work page 2003

[4] [3]

T. T. Deo, M. H. Bien, B. X. Hai, On division subrings norma lized by almost subnormal subgroups in division rings, Period. Math. Hung. (2019); DO I: 10.1007/s10998-019-00282-5

work page doi:10.1007/s10998-019-00282-5 2019

[5] [4]

T. H. Dung, A note on locally soluble almost subnormal sub groups in divsion rings, Int. J. Group Theory (2019), DOI: 10.22108/IJGT.2019.116399.154 6

work page arXiv 2019

[6] [5]

H. R. Dorbidi, R. Fallah-Moghaddam, M. Mahdavi-Hezaveh i, Soluble maximal subgroups in GLn(D), J. Algebra Appl. (6)10 (2011) 1371-1382

work page 2011

[7] [6]

Ebrahimian, Nilpotent maximal subgroups of GLn(D), J

R. Ebrahimian, Nilpotent maximal subgroups of GLn(D), J . Algebra 280 (2004) 244248

work page 2004

[8] [7]

Fallah-Moghaddam, Maximal subgroups of SL n(D), J

R. Fallah-Moghaddam, Maximal subgroups of SL n(D), J. Algebra 531 (2019) 70-82

work page 2019