On finitary linear groups with a locally 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.

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On locally finite groups with a locally nilpotent maximal subgroup

Archiv der Mathematik ◽

10.1007/bf01198716 ◽

1993 ◽

Vol 61 (3) ◽

pp. 215-220 ◽

Cited By ~ 3

Author(s):

Brunella Bruno ◽

Susan E. Schuur

Keyword(s):

Maximal Subgroup ◽

Finite Groups ◽

Locally Finite ◽

Locally Finite Groups ◽

Locally Nilpotent ◽

Nilpotent Maximal Subgroup

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On local-solvability of groups containing a locally nilpotent maximal subgroup

Archiv der Mathematik ◽

10.1007/bf01270607 ◽

1996 ◽

Vol 67 (6) ◽

pp. 441-447 ◽

Cited By ~ 2

Author(s):

Brunella Bruno ◽

Marianna Dalle Molle

Keyword(s):

Maximal Subgroup ◽

Local Solvability ◽

Locally Nilpotent ◽

Nilpotent Maximal Subgroup

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Irreducible locally nilpotent finitary skew linear groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500006209 ◽

1995 ◽

Vol 38 (1) ◽

pp. 63-76 ◽

Cited By ~ 7

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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