Local Systems

Local Systems of Locally Supersoluble Finitary Groups

Abstract

We show that locally supersoluble finitary linear groups over certain division rings (e.g. fields) have local systems of hypercyclic normal subgroups.

Background

Throughout the study of linear groups over fields and more generally, division rings, there has been effort to relate the property of local nilpotence (where every finitely generated subgroup is nilpotent) to hypercentrality (the group has a hypercentral series). For any group, hypercentrality implies local nilpotence. Garščuk [1] showed that the reverse is true for a linear group over a field.

Wehrfritz proved the supersoluble analogue of this result – a locally supersoluble linear group over a field is hypercyclic.

Zalesskii studied the nilpotence of linear groups over division rings and proved a partial result where there are certain conditions on the division ring (3). To date the general case has not been solved. 

Moving into the realm of finitary linear groups, there are counterexamples such as the McLain group which have a trivial centre and are locally nilpotent. Wehrfritz (2) proved that a locally nilpotent finitary group over a division ring has a local system of hypercentral normal subgroups. This paper examines the supersoluble analogue of this result and proves that a locally supersoluble finitary group over a division ring has a local system of hypercycle normal subgroups.

This work was undertaken as part of a PhD research project under the supervision of Prof B. A. F. Wehrfritz and funded by an EPSRC grant.

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REFERENCES
  1. M. S. Garaščuk, On the theory of generalized nilpotent linear groups, (Russian) Dokl. Akad. Nauk BSSR 4 (1960), 276-277.
  2. B. A. F. Wehrfritz, Nilpotence in finitary linear groups, Mich. Math. J. 40 (1993), 419-432.
  3. A. E. Zalesskii, The structure of several classes of matrix groups over a division ring, (Russian) Sibirsk. Math. Z 8(1967)1284-1298.