Free subgroups in maximal subgroups of GL_n(D)

ABSTRACT

Let D be a noncommutative finite dimensional F-central division algebra and M a noncommutative maximal subgroup of GL_n(D). It is shown that either M contains a noncyclic free subgroup or M is absolutely irreducible and there exists a unique maximal subfield K of M_n(D) such that K*M, K∕F is Galois with Gal(K∕F)≅M∕K* and Gal(K∕F) is a finite simple group.

KEYWORDS:

• Division ring
• free subgroup
• maximal subgroup

MATHEMATICS SUBJECT CLASSIFICATION:

• 16K20

Acknowledgments

The authors wish to thank the referee for useful comments. The first author thanks the Research Council of the University of Garmsar and the second author thanks the Sharif University of Technology for support.