# Sh:751

- Eda, K., & Shelah, S. (2002).
*The non-commutative Specker phenomenon in the uncountable case*. J. Algebra,**252**(1), 22–26. arXiv: math/0011231 DOI: 10.1016/S0021-8693(02)00045-5 MR: 1922382 -
Abstract:

An infinitary version of the notion of free products has been introduced and investigated by G.Higman. Let G_i (for i\in I) be groups and \ast_{i\in X} G_i the free product of G_i (i\in X) for X \Subset I and p _{XY}:\ast_{i\in Y} G_{i}\rightarrow \ast_{i\in X} G_{i} 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 (\ast_{i\in X} G_i, p_{XY}: X\subseteq Y\Subset I).We remark \ast_{i\in\emptyset} G_i=\{e\}. We prove:

Theorem: Let F be a free group. Then, for each homomorphism h: \lim \ast G_i \to F there exist countably complete ultrafilters u_0,\cdots,u_m on I such that h = h\cdot p_{U_0\cup \cdots\cup U_m} for every U_0\in u_0,\cdots ,U_m\in u_m.

If the cardinality of the index set I is less than the least measurable cardinal, then there exists a finite subset X_0 of I and a homomorphism \overline{h}:\ast _{i\in X_0}G_i\to F such that h=\overline{h}\cdot p_{X_0}, where p_{X_0}:\lim\ast G_i\to \ast_{i\in X_0}G_i is the canonical projection.

- published version (5p)

Bib entry

@article{Sh:751, author = {Eda, Katsuya and Shelah, Saharon}, title = {{The non-commutative Specker phenomenon in the uncountable case}}, journal = {J. Algebra}, fjournal = {Journal of Algebra}, volume = {252}, number = {1}, year = {2002}, pages = {22--26}, issn = {0021-8693}, doi = {10.1016/S0021-8693(02)00045-5}, mrclass = {20E06 (20K25)}, mrnumber = {1922382}, mrreviewer = {Paul C. Eklof}, doi = {10.1016/S0021-8693(02)00045-5}, note = {\href{https://arxiv.org/abs/math/0011231}{arXiv: math/0011231}}, arxiv_number = {math/0011231} }