From: "peter.neumann" 
Subject: Re: Simple groups in varieties of groups
Date: Thu, 14 Aug 1997 18:10:13 +0100 ()
 
Dear Arturo,
 
You write: 
 
> 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?"
 
You refer to the paper by G. A. Jones, `Varieties and simple groups' 
in J Australian Math Soc, XVII, 163--173  and ask:
 
> 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?
 
Yes, accepting that the Classification of the Finite Simple 
Groups is complete, Gareth's theorem answers my mother's problem 
because there are only finitely many more groups in the list than 
those that were known when he wrote his paper. 
 
All best wishes, $\Pi$eter
Queen's:  14.viii.97
____________________________________________________
 
Dr Peter M. Neumann, Queen's College, Oxford OX1 4AW 
tel. +44-1865-279 178 (messages: 279 120/21/22) 
fax: +44-1865-790 819
____________________________________________________


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Date: Mon, 18 Aug 1997 11:59:35 -0700 (PDT)
From: Arturo Magidin 
To: Group Pub Forum 
Subject: Re: Simple groups in varieties of groups

In response to my question on whether any proper subvariety of the variety
of all groups can have infinitely many nonisomorphic finite nonabelian
simple groups, Prof. Wiegold mentioned the existence some papers that
prove that the absolutely free group on two generators is residually in
any infinite collection of pairwise nonisomorphic finite nonabelian simple
groups.  Looking through MathSciNet (searching for "simple group*", "free
group*", "residually" and classification 20) yielded the following papers: 

Weigel, Thomas S.
Residual properties of free groups. 
J. Algebra 160 (1993), no. 1, 16--41. 

Weigel, Thomas S.
Residual properties of free groups. II. 
Comm. Algebra 20 (1992), no. 5, 1395--1425. 

Weigel, T. S.
Residual properties of free groups. III. 
Israel J. Math. 77 (1992), no. 1-2, 65--81. 

According to the reviews, these papers prove the result Prof. Wiegold
mentioned.

I appreciate all the help! Thanks again.

Regards,
Arturo Magidin
magidin@math.berkeley.edu