Some Completely Reducible Linear Groups over a Quaternion Division Ring Containing a Root Subgroup

2003 ◽  
Vol 31 (12) ◽  
pp. 5727-5754 ◽  
Author(s):  
Evgenii L. Bashkirov
Keyword(s):  
Division Ring ◽  
Linear Groups ◽  
Root Subgroup ◽  
Completely Reducible

scholarly journals Locally nilpotent skew linear groups

1986 ◽  
Vol 29 (1) ◽  
pp. 101-113 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Positive Integer ◽  
Division Ring ◽  
Matrix Ring ◽  
Linear Groups ◽  
Full Matrix ◽  
Locally Nilpotent ◽  
Completely Reducible ◽  
Full Matrix Ring

Throughout this paper D denotes a division ring with centre F and n a positive integer. A subgroup G of GL(n,D) is absolutely irreducible if the F-subalgebra F[G] enerated by G is the full matrix ring Dn ×n. It is completely reducible (resp. irreducible) if row n-space Dn over D is completely reducible (resp. irreducible), as D–G bimodule in the obvious way. Absolutely irreducible skew linear groups have a more restricted structure than irreducible skew linear groups, see for example [7],[8], [8] and [10]. Here we make a start on elucidating the structure of locally nilpotent suchgroups.

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.

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.

FINITARY AUTOMORPHISM GROUPS OVER NON-COMMUTATIVE RINGS

2001 ◽  
Vol 64 (3) ◽  
pp. 611-623 ◽  
Author(s):  
B. A. F. WEHRFRITZ
Keyword(s):  
Division Ring ◽  
Automorphism Groups ◽  
Commutative Rings ◽  
Linear Groups ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Finitary Automorphism

The notion of a group of finitary automorphisms of an arbitrary module over an arbitrary ring is introduced, and it is shown how properties of such groups can be derived from the case where the ring is a division ring (that is, from the properties of finitary skew linear groups). The results are particularly strong if either the group is locally finite or the module is Noetherian.

scholarly journals Coprime subdegrees for primitive permutation groups and completely reducible linear groups

2012 ◽  
Vol 195 (2) ◽  
pp. 745-772 ◽  
Author(s):  
Silvio Dolfi ◽  
Robert Guralnick ◽  
Cheryl E. Praeger ◽  
Pablo Spiga
Keyword(s):  
Permutation Groups ◽  
Linear Groups ◽  
Completely Reducible

scholarly journals Irreducible locally nilpotent finitary skew linear groups

1995 ◽  
Vol 38 (1) ◽  
pp. 63-76 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Linear Group ◽  
Division Ring ◽  
Linear Groups ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finitary Linear Group ◽  
Locally Nilpotent ◽  
Finitary Linear ◽  
Finite Dimensional Case

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dimDV(g-1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dimDV is infinite then dimDV = |G| = ℵ0 and G is a locally finite q-group for some prime q ≠ char D. Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.

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