M. Mahdavi-Hezavehi: Free Subgroups in Maximal Subgroups of $ GL_1(D)$

Mahdavih@sina.sharif.ac.ir

Submission: 2001, Jan 11

Let $ D $ be a division algebra of finite dimension over its centre $ F $. Given a noncommutative maximal subgroup $ M $ of $ D^* := GL_1(D) $, it is proved that either $ M $ contains a noncyclic free subgroup or there exists a maximal subfield $ K $ of $ D $ which is Galois over $ F $ such that $ K^* $ is normal in $ M $ and $ M/K^* \cong Gal(K/F) $. Using this result, it is shown in particular that if $ D $ is a noncrossed product division algebra, then $ M $ does not satisfy any group identity.

1991 Mathematics Subject Classification: 15A33, 16K

Keywords and Phrases: Free Subgroup, Division ring, maximal subgroup.

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