Date: Wed, 13 Aug 1997 16:15:59 -0700 (PDT)
From: Arturo Magidin 
To: 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