[xelxebar]

Thanks for taking the time to write something like this up. Your love of the subject comes across clearly, and it's a nice explanation!

My background is in algebra; these things excite me as well, but it's quite challenging to share the drama, beauty, and excitement with non-mathematicians.

Regarding the cyclic groups, they don't need to be constructed from a "trick" any more than the other finite groups. Think about the rotational symmetries of an n-sided polygon!

A lot of these finite simple groups have similar visualizations as symmetries of some geometric object. One of the tantalizing things about the monster group is that we haven't found anything like that for it. We don't know of what the monster group is a symmetry group.