Subgroups of division rings

Abstract

We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders. Our results are highly dependent on the computer software package Magma, although alternatives would probably work fine. A computer approach is required because our results require a listing of all maximal orders; the sheer number of maximal orders grows exponentially quickly and determining them all is not obtainable in polynomial time. When discriminants are large enough, the determination of a single maximal order is also exponentially hard. We indicate the limits as these problems arise.