This isn't exactly a homework problem-- it's on a sample exam. My first instinct is to look to matrix groups, since they are very often non-abelian and infinite, but I haven't had any luck.
asked Dec 14, 2011 at 20:56 836 2 2 gold badges 9 9 silver badges 17 17 bronze badges$\begingroup$ I think your instinct was right in considering matrix groups. Maybe you could check that the subgroup of invertible upper triangular matrices is solvable (start with $2$ by $2$ matrices to get the idea). $\endgroup$
Commented Dec 14, 2011 at 21:07 $\begingroup$ If you know dihedral groups, try the infinite dihedral group. $\endgroup$ Commented Dec 14, 2011 at 21:16 $\begingroup$ Thank you all. I'll look into these suggestions. $\endgroup$ Commented Dec 14, 2011 at 21:31$\begingroup$ @yoyo: Does that work? Or do you mean something other than unit norm quaternions? Those contain $SU(2)$, and that is simple, so hardly solvable? $\endgroup$
Commented Dec 14, 2011 at 21:38$\begingroup$ @yoyo: True, but my main point was that in which way will the unit quaternions form a solvable group? $\endgroup$