While I've got the Groupies' attention, let me bring up another puzzle that's been bothering me for 40 years or so. Can two non-isomorphic groups always be distinguished by some extrinsic property? For any group, we can find a number of derived objects: The order, the collection of element orders, the center, the commutator subgroup, the diagonal subgroup (generated by squares), &c. This process can be carried quite a ways, defining some pretty fancy concepts. Bill Henneman once explained the Frattini subgroup to me as "the set of useless generators". If you write out a notion like this in primitives, you are soon knee- deep in quantifiers. My question can be restated as: If two groups have enough of these concepts matching, are they isomorphic? Can the matchings be used to compute an isomorphism? For finite groups, I can check isomorphism in time roughly N^(log2(N)). The extrinsic properties might give a faster check. Similar questions exist for graphs & other math objects. Rich