Let T be a first order theory whose set of axioms is one of the standard finite sets of axioms for an
arbitrary group. Is it possible to have an axiomatizable first order extension of T which would 
characterize just those groups that are finite, simple and "sporadic"? In other words I am asking
whether the properties of being finite, simple and "sporadic" can each be expressed by a recursively
enumerable (or perhaps even finite) set of first order axioms? I realize, of course, that if we know
the "Monster" to be the largest such group and know its cardinal number, then our task becomes
considerably simpler. But I would consider this to be cheating and don't want to assume that we have
such knowledge. I confess that I don't know much about group theory and have never seen a clear
precise statement of what it means for a group to be "sporadic". Otherwise I would have refrained
from asking these questions which I suspect will all sound stupid to a group theorist.



A question about formulating first order axioms for group theory.
Because there are only finitely many sporadic simple groups, you can write, for each of them, a first-order sentence characterizing it up to isomorphism, and then you can form the disjunction of those sentences. But you don't want to build into your sentences our knowledge of the sporadic groups, and then you run into the theorem that, if a first-order theory has arbitrarily large finite models, then it also has infinite models. So any first-order characterization of the sporadic simple groups must depend on the fact that there are only finitely many of these groups. 


You also asked about characterizing, in a first-order way, the individual properties "finite", "simple", and "sporadic" (in contrast to just characterizing their conjunction). That is certainly impossible for "finite", by the theorem I quoted above. I would expect that it's also impossible for "simple" and "sporadic", but I don't know that for sure. 


