On the existence of normal maximal subgroups in division rings

Akbari, S.; Mahdavi-Hezavehi, M.

doi:10.1016/s0022-4049(01)00175-x

Cited by 12 publications

(12 citation statements)

References 5 publications

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“…But, when Z(D ) = 1 we have D * = F * × D . Hence, by [1,Theorem 6], F * has a normal maximal subgroup. So, D * has also a normal maximal subgroup.…”

Section: Lemmamentioning

confidence: 97%

On the non-triviality of G(D) and the existence of maximal subgroups of GL1(D
Keshavarzipour
¹
,
Mahdavi-Hezavehi
²

2005
Journal of Algebra
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Let D be an F -central division algebra of index n. Here we investigate a conjecture posed in [R. Hazrat et al., Reduced K-theory and the group G(D) = D * /F * D , in: Algebraic K-theory and its Applications, Trieste, 1997, pp. 403-409] that if D is not a quaternion algebra, then the group G 0 (D) = D * /F * D is non-trivial. Assume that either D is cyclic or F contains a primitive pth root of unity for some prime p | n. Using Merkurjev-Suslin Theorem, it is essentially shown that if none of the primary components of D is a quaternion algebra, then G(D) = D * /RN D/F (D * )D = 1. In this direction, we also study a conjecture posed in [S. Akbari, M. Mahdavi-Hezavehi, J. Pure Appl. Algebra 171 (2002) 123-131] or also [M. Mahdavi-Hezavehi, J. Algebra 271 (2004) 518-528] on the existence of maximal subgroups of D * .It is shown that if D is not a quaternion algebra with i(D) = p e , then D * has a maximal subgroup if either of the following conditions holds: (i) F has characteristic zero, or (ii) F has characteristic p, or (iii) F contains a primitive pth root of unity.  2004 Elsevier Inc. All rights reserved.

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“…But, when Z(D ) = 1 we have D * = F * × D . Hence, by [1,Theorem 6], F * has a normal maximal subgroup. So, D * has also a normal maximal subgroup.…”

Section: Lemmamentioning

confidence: 97%

On the non-triviality of G(D) and the existence of maximal subgroups of GL1(D
Keshavarzipour
¹
,
Mahdavi-Hezavehi
²

2005
Journal of Algebra
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“…In this paper we investigate some properties of maximal subgroups of the general skew linear group. The structure of such groups have been studied in various papers (e.g., see [1][2][3][4]). An interesting question which has not been answered yet is whether the multiplicative group of every noncommutative division ring has a maximal subgroup.…”

Section: Introductionmentioning

confidence: 99%

Nilpotent maximal subgroups of GLn(D)

Ebrahimian

2004

Journal of Algebra

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“…Let Nrd D : D * → F * be the reduced norm map, D (1) the kernel of this map and D the commutator subgroup of D * . The inclusion map F → D induces a homomorphism K 1 (F ) = F * → K 1 (D) = D * /D .…”

mentioning

confidence: 99%

“…Since x −n Nrd(x) ∈ D (1) and the reduced Whitehead group SK 1 (D) = D (1) /D is n-torsion (by [4], p. 157, Lemma 2), it follows that CK 1 (D) is an abelian group of bounded exponent n 2 . (In fact one can show that the bound is n, see the proof of Lemma 4, p. 154 in [4] or pp.…”

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confidence: 99%

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Nontriviality of certain quotients of K1 groups of division algebras

Hazrat

Wadsworth

2007

Journal of Algebra

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For a division algebra D finite dimensional over its center Z(D) = F , it is a conjecture that CK 1 (D) := Coker(K 1 F → K 1 D) is trivial if and only if D ∼ = ( −1,−1 F ) with F a formally real Pythagorean field. Since CK 1 (D) is very difficult to work with, we consider here NK 1 (D) := Nrd D (D * )/F * ind(D) , which is a homomorphic image of CK 1 (D). A field E is said to be NKNT if for every noncommutative division algebra D finite dimensional over E ⊆ Z(D), NK 1 (D) is nontrivial. It is proved that if E is finitely generated but not algebraic over some subfield then E is NKNT. As a consequence, if Z(D) is finitely generated over its prime subfield or over an algebraically closed field, then CK 1 (D) is nontrivial.

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On the existence of normal maximal subgroups in division rings

Cited by 12 publications

References 5 publications

On the non-triviality of G(D) and the existence of maximal subgroups of GL1(D
Keshavarzipour
¹
,
Mahdavi-Hezavehi
²

2005
Journal of Algebra
Self Cite
6
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0
0
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On the non-triviality of G(D) and the existence of maximal subgroups of GL1(D
Keshavarzipour
¹
,
Mahdavi-Hezavehi
²

2005
Journal of Algebra
Self Cite
6
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0
0
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Nilpotent maximal subgroups of GLn(D)

Nontriviality of certain quotients of K1 groups of division algebras

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On the existence of normal maximal subgroups in division rings

Cited by 12 publications

References 5 publications

On the non-triviality of G(D) and the existence of maximal subgroups of GL1(DKeshavarzipour1, Mahdavi-Hezavehi2 2005Journal of AlgebraSelf Cite6000View full textAdd to dashboardCite

On the non-triviality of G(D) and the existence of maximal subgroups of GL1(DKeshavarzipour1, Mahdavi-Hezavehi2 2005Journal of AlgebraSelf Cite6000View full textAdd to dashboardCite

Nilpotent maximal subgroups of GLn(D)

Nontriviality of certain quotients of K1 groups of division algebras

Contact Info

Product

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On the non-triviality of G(D) and the existence of maximal subgroups of GL1(D
Keshavarzipour
¹
,
Mahdavi-Hezavehi
²

2005
Journal of Algebra
Self Cite
6
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0
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On the non-triviality of G(D) and the existence of maximal subgroups of GL1(D
Keshavarzipour
¹
,
Mahdavi-Hezavehi
²

2005
Journal of Algebra
Self Cite
6
0
0
0
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