Date: Wed, 13 Aug 1997 16:15:59 -0700 (PDT) From: Arturo MagidinTo: Group Pub Forum Subject: Simple groups in varieties of groups Dear Group Pub, In Hanna Neumann's _Varieties of Groups_, she asks (Problem 23, pp. 166): "Can a variety other than O [the variety of all groups] contain an infinite number of non-isomorphic non-abelian finite simple groups?" In "Hanna Neumann's problems on varieties of groups" (Proc. Second Internat. Conf. Theory of Groups, Canberra 1973, pp.417-451), L.G. Kovacs and M.F. Newman say about this problem: "Heineken and Peter M. Neumann claimed, and Jones eventually proved, that no variety other than O contains infinitely many isomorphism types of the nonabelian finite simple groups which are known." Jones' paper also emphasizes that that other than the alternating and groups of Lie type "[...] there are only finitely many non-abelian simple groups known at present". He then proves that no variety other than O contains infinitely many isomorphism types of simple groups of alternating or Lie type. Given the current knowledge on finite simple groups, is Jones' theorem enough to fully answer Hanna Neumann's problem? Does any more work exist on this? I appreciate your help. Regards, Arturo Magidin magidin@math.berkeley.edu