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.

NILPOTENT AND LOCALLY FINITE MAXIMAL SUBGROUPS OF SKEW LINEAR GROUPS

2011 ◽  
Vol 10 (04) ◽  
pp. 615-622 ◽  
Author(s):  
M. RAMEZAN-NASSAB ◽  
D. KIANI
Keyword(s):  
Maximal Subgroup ◽  
Natural Number ◽  
Division Ring ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
Linear Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Matrix Groups ◽  
Nilpotent Maximal Subgroup

Let D be a division ring and N be a subnormal subgroup of D*. In this paper we prove that if M is a nilpotent maximal subgroup of N, then M′ is abelian. If, furthermore every element of M is algebraic over Z(D) and M′ ⊈ F* or M/Z(M) or M′ is finitely generated, then M is abelian. The second main result of this paper concerns the subgroups of matrix groups; assume D is a noncommutative division ring, n is a natural number, N is a subnormal subgroup of GLn(D), and M is a maximal subgroup of N. We show that if M is locally finite over Z(D)*, then M is either absolutely irreducible or abelian.

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.

Free subgroups in maximal subgroups of SLn(D)

2020 ◽  
pp. 2150071
Author(s):  
R. Fallah-Moghaddam
Keyword(s):  
Division Ring ◽  
Finite Simple Group ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Central Division ◽  
Normal Abelian Subgroup ◽  
Finite Dimensional ◽  
Free Subgroups

Given a non-commutative finite-dimensional [Formula: see text]-central division ring [Formula: see text], [Formula: see text] a subnormal subgroup of [Formula: see text] and [Formula: see text] a non-abelian maximal subgroup of [Formula: see text], then either [Formula: see text] contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a subfield of [Formula: see text], [Formula: see text] is Galois and [Formula: see text], also [Formula: see text] is a finite simple group with [Formula: see text].

scholarly journals Free subgroups in maximal subgroups of skew linear groups

2019 ◽  
Vol 29 (03) ◽  
pp. 603-614 ◽  
Author(s):  
Bui Xuan Hai ◽  
Huynh Viet Khanh
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
General Linear Group ◽  
Subnormal Subgroup ◽  
Free Subgroup ◽  
Maximal Subgroups ◽  
Linear Groups ◽  
Locally Finite ◽  
Starting Point ◽  
Free Subgroups

The study of the existence of free groups in skew linear groups have begun since the last decades of the 20th century. The starting point is the theorem of Tits (1972), now often referred to as Tits’ Alternative, stating that every finitely generated subgroup of the general linear group [Formula: see text] over a field [Formula: see text] either contains a non-cyclic free subgroup or it is solvable-by-finite. In this paper, we study the existence of non-cyclic free subgroups in maximal subgroups of an almost subnormal subgroup of the general skew linear group over a locally finite division ring.

SOLUBLE MAXIMAL SUBGROUPS IN GLn(D)

2011 ◽  
Vol 10 (06) ◽  
pp. 1371-1382 ◽  
Author(s):  
H. R. DORBIDI ◽  
R. FALLAH-MOGHADDAM ◽  
M. MAHDAVI-HEZAVEHI
Keyword(s):  
Maximal Subgroup ◽  
Division Ring ◽  
Real Closed Field ◽  
Maximal Subgroups ◽  
Closed Field

Let D be an F-central non-commutative division ring. Here, it is proved that if GL n(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F] < ∞, and D is cyclic of degree p, a prime. Furthermore, a classification of soluble maximal subgroups of GL n(F) for an algebraically closed or real closed field F is also presented. We then determine all soluble maximal subgroups of GL 2(F) for fields F with Char F ≠ 2.

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.

POLYNOMIAL AND GROUP IDENTITIES IN ALTERNATIVE LOOP ALGEBRAS

2008 ◽  
Vol 07 (05) ◽  
pp. 593-599
Author(s):  
EDGAR G. GOODAIRE ◽  
CÉSAR POLCINO MILIES
Keyword(s):  
Polynomial Identity ◽  
Group Identity ◽  
Loop Algebra ◽  
Polynomial Identities ◽  
Loop Algebras ◽  
Group Identities ◽  
Loop Ring

We investigate polynomial identities on an alternative loop algebra and group identities on its (Moufang) unit loop. An alternative loop ring always satisfies a polynomial identity, whereas whether or not a unit loop satisfies a group identity depends on factors such as characteristic and centrality of certain kinds of idempotents.

On Conjugates in Division Rings

1958 ◽  
Vol 10 ◽  
pp. 374-380 ◽  
Author(s):  
Carl C. Faith
Keyword(s):  
Division Algebra ◽  
Division Ring ◽  
Galois Theory ◽  
Division Rings ◽  
Finite Dimensional ◽  
Inner Automorphism ◽  
Elementary Methods

Let D be a non-commutative division ring with centre C, and let Δ be a proper division subring not contained in C. In (4) Cartan raised the question: is it ever possible for each inner automorphism of D to induce an automorphism of Δ? As is well-known, Cartan (4, Théorème 4) with the aid of his Galois Theory answered this negatively in case D is a finite dimensional division algebra. Later Brauer (3), and Hua (8), using elegant, elementary methods, extended Cartan's theorem to arbitrary division rings.

A NOTE ON GROUP IDENTITIES IN DIVISION RINGS

2004 ◽  
Vol 47 (3) ◽  
pp. 557-560 ◽  
Author(s):  
M. A. Chebotar ◽  
P.-H. Lee
Keyword(s):  
Division Ring ◽  
Group Identity ◽  
Subject Classification ◽  
Division Rings ◽  
Group Of Units ◽  
Trivial Group ◽  
Group Identities

AbstractLet $D$ be a division ring whose group of units satisfies a non-trivial group identity $w$. Let $\alpha$ be the sum of positive degrees of indeterminates occurring in $w$. If the centre of $D$ contains more than $3\alpha$ elements, then $D$ is commutative.AMS 2000 Mathematics subject classification: Primary 16R50. Secondary 16K20

scholarly journals Maximal subgroups of finite groups

1990 ◽  
Vol 13 (2) ◽  
pp. 311-314
Author(s):  
S. Srinivasan
Keyword(s):  
Maximal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Fitting Subgroup ◽  
Small Index

In finite groups maximal subgroups play a very important role. Results in the literature show that if the maximal subgroup has a very small index in the whole group then it influences the structure of the group itself. In this paper we study the case when the index of the maximal subgroups of the groups have a special type of relation with the Fitting subgroup of the group.

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