[doonesbury]

When I wrote:

"then you can do line symmetry of (a+b)+c through c and on and on an on and on ... until at some point you've played through all the symmetry on all the permutation and choices of elements until it goes back into something you started with earlier. Extending the analogy monster groups have rotation, translate, mirror, and zillion other kinds of symmetries ..."

That's slightly misleading. What I should have wrote is starting with two random elements in the set a,b form c=a+b. Almost always this will yield a new, distinct element in the set c. So throw c into the set and repeat trying all permutations and combinations getting d,e,f,...z.

At some point the symmetry in the monster group will overtake insuring that, because of symmetry, a subsequent operation on z will transform it equal an earlier element. Thus the monster group maintains an interesting ultimate boundary where all elements eventually through symmetry revert back to an earlier form.

In the article's 90deg rotation analogy on 2D squares, we can rotate a to b to c thus making 3 distinct squares (if we label their corners), however, upon rotating c again we get back to a. In monster groups it takes a longer run to revert.

I suppose it's worth emphasizing monster groups of finite size can only exist in a few variations. And between two monster groups M,O there is no in-between group N. It's definitely discrete in that sense.