From: peter.neumann@queens.ox.ac.uk To: Arturo MagidinSubject: Re: number of varieties Date: Sat, 19 Apr 1997 15:14:44 +0100 () On Fri, 18 Apr 1997 10:40:03 -0700 (PDT) Arturo Magidin wrote: > Dear Group Pub, > > Does anyone know how many varieties of groups of finite exponent there > are? Are there uncountably many, or just countably many? > > Thanks. Regards, > Arturo Magidin > magidin@math.berkeley.edu > Ol'shanski in Izvestia. Akad. Nauk SSSR 1970 produced $2^{\aleph_0}$ varieties of exponent $8pq$ where $p, q$ are distinct odd prime numbers. Vaughan-Lee in Bull. London Math. Soc. 1970 produced $2^{\aleph_0}$ varieties of exponent $16$. There have been many refinements since. All best wishes, $\Pi$eter Queen's: 19.iv.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: Sun, 20 Apr 1997 13:09:16 +1000 From: "M.F.(Mike) Newman" Subject: Re: number of varieties For anyone who wants to follow more recent work, start at: [1] 97e:20038 Gupta, C. K.; Krasil\cprime nikov, A. N. Metanilpotent varieties without torsion and varieties of groups of prime power exponent. Internat. J. Algebra Comput. 6 (1996), no. 3, 325--338. (Reviewer: A. I. Budkin) 20E10 (I used this to test mathscinet with: Matches for: Anywhere=(group) AND Anywhere=(variety) AND Anywhere=(torsion) which gave 422 replies many irrelevant but this is the first, i.e. latest) Mike Newman