On a recent theorem of M. Shirvani on subgroups of division algebras

In some very nice, recent and yet to be published work Shirvani produces some very tight bounds for the index of an abelian normal subgroup in a soluble-by-finite subgroup of a finite-dimensional division algebra. Here we present an alternative proof of Shirvaniʹs Theorem. With a little more work our method also yields slightly improved bounds. Finally we present examples that restrict further improvements to the bounds and discuss a couple of variants.