Even order subgroups of finite dimensional division rings Public Deposited

http://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/fn107321f

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  • Let K be a field, and G a finite group. G is said to be K-adequate if there exists a division ring D, finite dimensional over K, and with center K, such that G is contained in the multiplicative group of nonzero elements of D. In this dissertation we investigate the notion of K-adequacy under the assumptions that K is an algebraic number field or p-local field and G is a noncyclic group of even-order. Results in this area depend upon the classification of K-division rings by means of Hasse invariants, and Amitsur's classification of those finite groups which can be embedded in the multiplicative group of some division ring. It is shown that if K is an algebraic number field then there exists a noncyclic group of even-order which is K-adequate. We show this is not true if K is a p-local field and determine necessary and sufficient conditions on K for there to exist a noncyclic group of even-order which is K-adequate. Combining this with previous work on noncyclic odd-order subgroups we determine when the restriction that the noncyclic group be of even-order may be dropped.
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