From: bikenaga@bellatlantic.net (Bruce Ikenaga) Subject: Re: Infinite group Date: Thu, 11 Jan 2001 04:02:50 GMT Newsgroups: sci.math Summary: Infinite groups all of whose proper subgroups are finite. On 10 Jan 2001 10:37:20 GMT, mareg@primrose.csv.warwick.ac.uk () wrote: >In article <93f616$lvr$1@nnrp1.deja.com>, bill_pet@my-deja.com writes: >>Is there a non abelian exmaple for such group ? >> >>Thanks, >>Bill >> >> >>In article <93cr3k$u6a$2@uwm.edu>, >> David G Radcliffe wrote: >>> ahmedfares@my-deja.com wrote: >>> : Is there an infinite group G with no infinite proper subgroups ? >>> : i.e all proper subgroups of G are finite ? > > >Yes, there are such examples, but the only ones that I know of are the >so-called Tarski Monsters. For large enough primes p, there are infinite >groups in which all proper subgroups are finite of order p. >These were proved to exist in the 1970's (I think by Rips initially), and the >proofs are extremely long and difficult. > >Any example of the sort you are looking for woudl have to be a torsion group >(all elements of finite order). I suspect that it would need to be finitely >generated, although I can't prove that immediately! A. Yu. Olshanskii (An infinite group with subgroups of prime orders, Math. USSR Izvestija, 16(1981), no. 2, 279--289) constructs an infinite nonabelian group all of whose proper subgroups have prime order. Any two subgroups of the same order are conjugate, there's a presentation with two generators, and the word and conjugacy problems are solvable. In another paper, he shows that one can construct for each sufficiently large prime p an infinite group in which every proper subgroup has order p. (Has anyone written an exposition of this diagram method stuff? The original papers look like really tough sledding!) ============================================================================== From: mareg@mimosa.csv.warwick.ac.uk () Subject: Re: Infinite group Date: 11 Jan 2001 09:17:58 GMT Newsgroups: sci.math In article <3a5d2dec.66047205@news.bellatlantic.net>, bikenaga@bellatlantic.net (Bruce Ikenaga) writes: [Above article was quoted --djr] Right. I first heard of the existence of these groups in a seminar given by Rips in the late 1970's, but the first published proof was indeed by Olshanskii. Avinoam Mann has pointed out to me in an e-mail that, by a theorem of P.Hall, Kulatilaka and Kargapolov, an infinite locally finite group contains an infinite abelian subgroup. This implies that any nonabelian infinite group with all of its proper subgroups finite would be finitely generated. (Localy finite means all finitely generated subgroups finite.) Derek Holt.