The classification of finite simple groups states roughly that every non-abelian finite simple group is either alternating, a group of Lie type, or a sporadic group.
For each of the groups of Lie type and alternating groups, there are infinite versions. Roughly speaking, there is an underlying object (a set or a field) and there is no problem with letting that object take infinite proportions. Actually this suggests a wealth of other groups (forms of the groups of Lie type) of which only a small number is realized over finite fields (for instance because there are not many quadratic forms over finite fields).
I was wondering whether there are indications that some sporadic groups can be defined in terms of fields (or maybe graphs) with very special properties which ensure that there are is only one finite family member, although there could be a lot of infinite ones.
For instance, Mathieu groups can be defined as automorphism groups of certain Steiner systems and conceivably there is a natural axiomatization of such Steiner systems which also produces infinite members.
In a similar vein, I learned in the answer to this question, that the sporadic group $Fi_{22}$ arises from a geometry that is closely related to that of $U_6(2)$. Certainly there are many similar related results, relating a sporadic groups $S$ with a groups of Lie type $L(\mathbb F_q)$. Conceivably, at least in one case it could appear that the construction which produces $S$ out of $L(\mathbb F_q)$ generalizes naturally to, say, extensions $k/\mathbb F_q$ without algebraic subextension, so that there are groups $S(k)$ constructed out of $L(k)$ for any such $k$, even though there is still only one finite such group.
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I just found the following text on https://ncatlab.org/nlab/show/Monster+group which seems relevant to my question.
There is a school of thought, going back to at least Israel Gelfand, that sporadic groups are really members of some other infinite families of algebraic objects, but due to numerical coincidences or the like, just happen to be groups (see this nCafe post). One version of this, in the case of the Monster (and perhaps for other sporadic groups via Moonshine phenomena) is that what we know as the Monster is just a shadow of a 2-group, as the Monster can be constructed as an automorphism group of a conformal field theory, a structure rich enough to have a automorphism 2-group(oid) (see this nCafe discussion).
