Re: Homomorphisms on Z^n

["Stephen Urban Chase" <suc1@cornell.edu>]

Zeeman proved the assertion in 1955 for non-measurable cardinals n.  There
are more general results which comprise the theory of slender (abelian)
groups; see, e.g., Chapter XIII, Sections 94-95 of [L. Fuchs, Infinite
Abelian Groups vol. 2, Academic Press, 1973], especially Corollary 94.6 on
p 162.

I haven't thought much about abelian groups since the early 1960's, but it
is well known that infinite direct products have very interesting
properties, both as abstract and as topological groups.  For example, a
closed subgroup of a direct product of countably many copies of  Z  is
also a direct product, but not so for uncountable products (see [R.J.
Nunke, On direct products of infinite cyclic groups, Proc. Amer. Math.
Soc. 13 (1962), pp 66-71]).  In fact, Zeeman's result implies that a
countable free group is a closed subgroup of a direct product with
uncountably many factors.  A generalization of Nunke's theorem and some
related results are contained in my old paper [Function topologies on
abelian groups, Ill. J. Math. 7 (1963), pp 593-608].

Steve Chase



---------------------------- Original Message ----------------------------
Subject: categories: Homomorphisms on Z^n
From:    "Michael Barr" <barr@math.mcgill.ca>
Date:    Fri, September 14, 2007 8:34 am
To:      "Categories list" <categories@mta.ca>
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Many years ago (at least 45) Harrison mentioned to me that for any n
(including infinite cardinals), Hom(Z^n,Z) = n.Z, in other words the
Z-dual of the product is the sum.  This is obviously a very special
property of Z, almost the negation of injectivity.  Has anyone on this
list ever seen this before and can give me a reference?

Michael