From: magidin@bosco.berkeley.edu (Arturo Magidin) Subject: Re: homomorphic images of products of a group Date: 19 Nov 1999 19:15:48 GMT Newsgroups: sci.math.research Keywords: varieties of groups In article <377CF6E4.1EEEE3D9x@timesup.org>, Timmy B wrote: >I am wondering whether anyone knows of concrete results about >H P (G), the set of homomorphic images of direct products of G. >For instance, in the abelian case it is the case that H(G) = S(G), >i.e. all homomorphic images are subgroups and all subgroups are >homomorphic images, so H P (G) = H S P (G), the variety of G. Careful! H(G) is not in general equal to S(G), even for finitely generated abelian groups. Consider the abelian quasicylic group G Z_{p^\infty}; this can be constructed as the subgroup (under addition) of Q/Z of all fracions that can be written as b/p^a for some a>0. Every subgroup of G is cyclic, and every cyclic group of order a power of p is a subgroup of G. However, every quotient of G is either trivial or isomorphic to G. So S(G) are all cylic groups of order a power of p (including the trivial group), plus G, whereas H(G) is only G and the trivial group. For a finitely generated example, consider the infine cyclic group Z. Then H(Z) is the collection of all cyclic groups (finite or infinite), whereas every subgroup of G is either infinite cyclic or trivial, so S(G) is only the trivial and the infinite cyclic groups. I think S(G)=H(G) holds for FINITE abelian groups, though. It should be enough to show it for finite abelian groups of prime power order. >Cam one say anything else about this situation? The operators H,S,P generate an ordered semigroup with at most 18 elements. Several things can be said, and you can find a diagram of the resulting lattice in sundry places. Look at: Bergman, George M. "HSP \not= SHPS for Metabelian groups and related results", Algebra Universalis 26 (1989) no. 3 pp.267-283 which has a diagram and some references regarding this lattice. Check also Neumann, P.M. "The inequality of SQPS and QSP as operators on classes of groups", Bull. Amer. Math. Soc. 76 (1970) pp. 1067-1069. ====================================================================== "It's not denial. I'm just very selective about what I accept as reality." --- Calvin ("Calvin and Hobbes") ====================================================================== Arturo Magidin magidin@math.berkeley.edu ============================================================================== From: magidin@bosco.berkeley.edu (Arturo Magidin) Subject: Re: homomorphic images of products of a group Date: 22 Nov 1999 17:57:34 GMT Newsgroups: sci.math.research In article <3837E585.103D8907@timesup.org>, Timmy B wrote: >Yes, thanks. I should have put in that I am, in general, interested in >finite examples - something I almost always mean, and almost always >forget to write in. But the infinite examples are interesting, thank >you. So, if you are interested in finite examples, are you considering the operator P (arbitrary direct products), or the opreator P_f (direct products with finitely many factors)? For a finite group, one can show the analog of Birkhoff's theorem and prove that if you have a variety generated by finitely many finite groups, then the finitely generated groups in that variety are homomorphic images of subgroups of ->finite<- direct powers of the generating groups (and if you have a variety generated by finitely many finite groups then it is locally finite, which is now an easy consequence of this result). This is Higman's "finite HSP", and can be found at: Graham Higman, _Some remarks on varieties of groups_ Quart J. Math. Oxford (2) 10 (1959) pp. 165-178. You might also want to look in Hanna Neumann's _Varieties of Groups_ (Ergebnisse der Mathematik und ihrer Grenzgebiete New Series Vol. 37, Springer-Verlag), which has the state of the art up to 1967 (of course, lots of things happened since then, but still). Although I don't have it in front of me right now, but I seem to remember a discussion on locally finite varieties. >> I think S(G)=H(G) holds for FINITE abelian groups, though. It should >> be enough to show it for finite abelian groups of prime power order. >> >> >Cam one say anything else about this situation? >> >> The operators H,S,P generate an ordered semigroup with at most 18 >> elements. Several things can be said, and you can find a diagram of >> the resulting lattice in sundry places. > >I will do. Can anyone suggest a place where some results for these >operators on finite groups might be found? FOr instance, when (other >than for finite abelian groups) is HSP = HP? It's actually a fairly strong condition, which you can see by looking at the positions in the lattice of HP and HSP. Bergman's paper (which I cited in my previous response) has a number of references you might want to check out about the operators, and results on when you get equalities and when you don't and so on; the same is true of Peter Neumann's paper, which I also cited in my previous response. Although many of those papers deal with the operators from the point of view of General Algebra, you'll find examples on groups scattered throughout. (By the way, Bergman's paper has a diagram of the lattice generated by the operators H,S, and P; a similar lattice is obtained if you replace P by P_f). ====================================================================== "It's not denial. I'm just very selective about what I accept as reality." --- Calvin ("Calvin and Hobbes") ====================================================================== Arturo Magidin magidin@math.berkeley.edu