Periodic Linear Groups with the Weak Chain Conditions on Subgroups of Infinite Central Dimension

Abstract

Let V be a vector space over a field F. If G ≤ GL(V, F), the central dimension of G is the F-dimension of the vector space V/C _V (G). In Dixon et al. (2004) and Kurdachenko and Subbotin (2006), soluble linear groups in which the set ℒ _icd(G) of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. In this article we study periodic locally radical linear groups, in which the set ℒ _icd(G) satisfies one of the weak chain conditions: the weak minimal condition or the weak maximal condition.

Key Words:

• Central dimension of a linear group
• Linear group
• Radical group
• Weak maximal condition
• Weak minimal condition

Mathematics Subject Classification:

• Primary 20F22
• Secondary 20H20

ACKNOWLEDGMENT

The authors wish to thank the referee for his/her excellent work and useful comments that have allowed to improve the layout and the contents of this article.

This article is supported by Proyecto MTM2004-04842 of Dirección General de Investigación MEC (Spain).

Notes

Communicated by M. R. Dixon