The non-commutative Specker phenomenon in the uncountable case
Abstract
An infinitary version of the notion of free products has been introduced and investigated by G.Higman. Let Gi (for i in I) be groups and asti in X Gi the free product of Gi (i in X) for X Subset I and pXY: asti in Y Gi->asti in X Gi the canonical homomorphism for X subseteq Y Subset I. (X Subset I denotes that X is a finite subset of I.) Then, the unrestricted free product is the inverse limit lim (asti in X Gi, pXY: X subseteq Y Subset I). We remark asti in emptyset Gi= e . We prove: Theorem: Let F be a free group. Then, for each homomorphism h:lim ast Gi-> F there exist countably complete ultrafilters u0,...,um on I such that h = h . pU0 cup ... cup Um for every U0 in u0, ...,Um in um. If the cardinality of the index set I is less than the least measurable cardinal, then there exists a finite subset X0 of I and a homomorphism overline h: asti in X0Gi-> F such that h= overline h . pX0, where pX0: lim ast Gi->asti in X0Gi is the canonical projection.
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