Maximal Subgroups of Subnormal Subgroups of GL_n(D) with Finite Conjugacy Classes

Abstract

Let D be an infinite division ring. A famous result due to Herstein says that every noncentral element of D has infinitely many conjugates in D. So, if D* is an FC-group, then D is a field. Now let N be a subnormal subgroup of GL _n (D), where n ≥ 1, and M a maximal subgroup of N. In this article, we prove that if M is an FC-group, then M is contained in the multiplicative group of some subfield of M _n (D).

Key Words:

• Division ring
• FC-group
• Maximal subgroup
• Subnormal sub-group

2000 Mathematics Subject Classification:

• 20H25
• 15A33

ACKNOWLEDGMENTS

The authors indebted to the referee for detailed suggestions which considerably improved the readability of this article. The research of the first author was in part supported by a grant from IPM with number (No. 88050116).

Notes

Communicated by M. Bresar.