Return to Revisions

This addresses the second question "What is known about finite subgroups of $SU(n)$". A special case of the Margulis lemma implies that for each $n$, there is an $m(n)$ such that any finite subgroup of $O(n)$ has an abelian subgroup of index $m(n)$ (see Corollary 4.2.4 of Thurston's book). Thus, there is a normal abelian subgroup of index at most $m!$. So one may make a statement: there are finitely many finite groups so that any finite subgroup of $SU(n)$ is an abelian extension (of rank at most $n-1$) of one of these finitely many groups. It would be quite interesting to obtain an estimate of the function $m(n)$, which should be possible by giving an effective proof of Margulis' theorem. I did a literature search once to see if anyone had attempted this, but I didn't find anything, and I would curious if someone knows something.