On Locally Nilpotent Subgroups of GL₁(D)

Abstract

In this article we study locally nilpotent subgroups of D*: = GL ₁(D), where D is a division ring. It is proved that every locally nilpotent subnormal subgroup of D* is central. If D is algebraic over its centre then every locally solvable subnormal subgroup of D* is central. Also, in this case, it is shown that every locally nilpotent maximal subgroup of D* can occur as the multiplicative group of some maximal subfield of D.

Key Words:

• Algebraic division rings
• Linear groups
• Locally nilpotent
• Maximal

2000 Mathematics Subject Classification:

• 16K20

ACKNOWLEDGMENT

The authors thank the referee for his or her comments and suggestions.

Notes

Communicated by M. R. Dixon.