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 ____________________________________________________ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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