scholarly journals THE DERIVED LENGTH OF A SOLUBLE SUBGROUP OF A FINITE-DIMENSIONAL DIVISION ALGEBRA

2006 ◽  
Vol 48 (01) ◽  
pp. 119 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Division Algebra ◽  
Derived Length ◽  
Finite Dimensional ◽  
Soluble Subgroup

scholarly journals Some matrix groups over finite-dimensional division algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 97-111 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Division Algebra ◽  
Finite Dimension ◽  
Linear Groups ◽  
Matrix Groups ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Skew Fields ◽  
Rank 2 ◽  
Soluble Subgroup ◽  
Free Subgroups

Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

Existence of Nonabelian Free Subgroups in the Maximal Subgroups of GLn(D)

2014 ◽  
Vol 21 (03) ◽  
pp. 483-496 ◽  
Author(s):  
H. R. Dorbidi ◽  
R. Fallah-Moghaddam ◽  
M. Mahdavi-Hezavehi
Keyword(s):  
Maximal Subgroup ◽  
Division Algebra ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Central Division ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Free Subgroups ◽  
Maximal Subfield

Given a non-commutative finite dimensional F-central division algebra D, we study conditions under which every non-abelian maximal subgroup M of GLn(D) contains a non-cyclic free subgroup. In general, it is shown that either M contains a non-cyclic free subgroup or there exists a unique maximal subfield K of Mn(D) such that NGLn(D)(K*)=M, K* ◁ M, K/F is Galois with Gal (K/F) ≅ M/K*, and F[M]=Mn(D). In particular, when F is global or local, it is proved that if ([D:F], Char (F))=1, then every non-abelian maximal subgroup of GL1(D) contains a non-cyclic free subgroup. Furthermore, it is also shown that GLn(F) contains no solvable maximal subgroups provided that F is local or global and n ≥ 5.

scholarly journals Nicely semiramified division algebras over Henselian fields

2005 ◽  
Vol 2005 (4) ◽  
pp. 571-577 ◽  
Author(s):  
Karim Mounirh
Keyword(s):  
Division Algebra ◽  
Division Algebras ◽  
Central Division ◽  
Field Extensions ◽  
Finite Dimensional ◽  
Central Division Algebra ◽  
Radical Type ◽  
Maximal Subfield

This paper deals with the structure of nicely semiramified valued division algebras. We prove that any defectless finite-dimensional central division algebra over a Henselian fieldEwith an inertial maximal subfield and a totally ramified maximal subfield (not necessarily of radical type) (resp., split by inertial and totally ramified field extensions ofE) is nicely semiramified.

scholarly journals The Norm of a Skew Polynomial

2021 ◽  
Author(s):  
S. Pumplün ◽  
D. Thompson
Keyword(s):  
Division Algebra ◽  
Polynomial Ring ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Skew Polynomial Ring ◽  
Skew Polynomials ◽  
Finite Dimensional ◽  
Central Simple ◽  
Skew Polynomial ◽  
Reduced Norm

AbstractLet D be a finite-dimensional division algebra over its center and R = D[t;σ,δ] a skew polynomial ring. Under certain assumptions on δ and σ, the ring of central quotients D(t;σ,δ) = {f/g|f ∈ D[t;σ,δ],g ∈ C(D[t;σ,δ])} of D[t;σ,δ] is a central simple algebra with reduced norm N. We calculate the norm N(f) for some skew polynomials f ∈ R and investigate when and how the reducibility of N(f) reflects the reducibility of f.

scholarly journals Identities on Maximal Subgroups of GLn(D)

2005 ◽  
Vol 12 (03) ◽  
pp. 461-470 ◽  
Author(s):  
D. Kiani ◽  
M. Mahdavi-Hezavehi
Keyword(s):  
Maximal Subgroup ◽  
Division Algebra ◽  
Division Ring ◽  
Group Identity ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
Polynomial Identities ◽  
Linear Hull ◽  
Finite Dimensional ◽  
Group Identities

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n≥1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]<∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M=K* where K is a field and [D:F]<∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GLn(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite.

Intrinsic Functions on Matrices of Real Quaternions

1963 ◽  
Vol 15 ◽  
pp. 456-466 ◽  
Author(s):  
C. G. Cullen
Keyword(s):  
Division Algebra ◽  
Matrix Algebra ◽  
Simple Algebra ◽  
Real Field ◽  
Complex Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Simple Algebras ◽  
Real Quaternions ◽  
Algebra Over A Field

It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.

On Involutions of Quasi-Division Algebras

1975 ◽  
Vol 17 (5) ◽  
pp. 723-725 ◽  
Author(s):  
Lowell Sweet
Keyword(s):  
Division Algebra ◽  
Short Paper ◽  
Division Algebras ◽  
Finite Dimensional

All algebras are assumed to be finite dimensional and not necessarily associative. An involution of an algebra is an algebra automorphism of order two. A quasi-division algebra is any algebra in which the non-zero elements form a quasi-group under multiplication. The purpose of this short paper is to determine the structure of all involutions of quasi-division algebras and to give an application of this result.

scholarly journals VARIATIONS SUR UN THÈME DE ALDAMA ET SHELAH

2016 ◽  
Vol 81 (1) ◽  
pp. 96-126
Author(s):  
CÉDRIC MILLIET
Keyword(s):  
Elementary Extension ◽  
Independence Property ◽  
Derived Length ◽  
Definable Subset ◽  
Soluble Subgroup

AbstractWe consider a group G that does not have the independence property and study the definability of certain subgroups of G, using parameters from a fixed elementary extension G of G. If X is a definable subset of G, its trace on G is called an externally definable subset. If H is a definable subgroup of G, we call its trace on G an external subgroup. We show the following. For any subset A of G and any external subgroup H of G, the centraliser of A, the A-core of H and the iterated centres of H are external subgroups. The normaliser of H and the iterated centralisers of A are externally definable. A soluble subgroup S of derived length ℓ is contained in an S-invariant externally definable soluble subgroup of G of derived length ℓ. The subgroup S is also contained in an externally definable subgroup X ∩ G of G such that X generates a soluble subgroup of G of derived length ℓ. Analogue results are discussed when G is merely a type definable group in a structure that does not have the independence property.

scholarly journals Commuting rings of simple A(k)-modules

1981 ◽  
Vol 31 (2) ◽  
pp. 142-145 ◽  
Author(s):  
Daniel R. Farkas ◽  
Robert L. Snider
Keyword(s):  
Division Algebra ◽  
Division Ring ◽  
Characteristic Zero ◽  
Weyl Algebra ◽  
Division Algebras ◽  
Finite Dimensional ◽  
Algebra Over A Field

AbstractFor the Weyl algebra A(k) and each finite dimensional division ring D over k, there exists a simple A(k)-module whose commuting ring is D.It has been known for some time that if A(k) denotes the Weyl algebra over a field k of characteristic zero, the commuting ring of a simple A(k)-module is a division algebra finite dimensional over k (see the introduction of [1]). Which division algebras actually appear? Quebbemann [1] showed that if D is a finite dimensional division algebra whose center is k, then it occurs as a commuting ring. We complete this circle of ideas by showing that any D appears: a division algebra over k appears as the commuting ring of a simple A(k)-module if and only if it is finite dimensional over k.

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