Perhaps they don’t.

No, I’m not questioning whether the Mathieu groups or the Monster actually exist. They do.

What I’d like to highlight is that “sporadic” is not a property of a group, the way “commutative” or “simple” or “nilpotent” are. “Sporadic” is an indication of our organizational structure of finite simple groups, and our organizational structure may just be incomplete or ignorant.

Perhaps there’s a unifying way of thinking of all the finite simple groups which turns “sporadic” into “natural and fitting in the grand scheme of things”. We may just not have found the appropriate unifying language or structure.

At the same time, maybe they really are sporadic. Maybe there’s no unifying underlying scheme and there really are prime cyclic, alternating, Lie-type and 26 weird, special, inexplicable finite simple groups.

If this is so, why are they there? I don’t know that this is a meaningful question. They just are. You can think of this as an instance of the “law of small numbers”: every finite simple group is in one of the big families, except that for small numbers there are some peculiar coincidences that don’t recur.

There are many instances of this “law” (really, just a whimsical observation) throughout math. All prime numbers are odd, except for one weird outlier. Why does [math]2[/math] exist? Well, someone has to divide all of those even numbers. Why? Because. It’s a necessary consequence of the idea of multiplication of natural numbers.

All regular polyhedra, in all dimensions, are (hyper)cubes, simplices and cross polytopes. Except, in dimension 2 you have infinitely many and in dimension 3 we have two extra special ones: the icosahedron and the dodecahedron. Why do they exist? They do. Small dimensions impose fewer requirements and enable more lucky combinations.

That’s how I see this.