Sh:302a

• Grossberg, R. P., & Shelah, S. (1998). On cardinalities in quotients of inverse limits of groups. Math. Japon., 47(2), 189–197. arXiv: math/9911225 MR: 1615081
• Abstract:
  Let \lambda be \aleph_0 or a strong limit of cofinality \aleph_0. Suppose that \langle G_m,\pi_{m,n}\;:\;m\leq n<\omega\rangle and \langle H_m,\pi^t_{m,n}\;:\;m\leq n<\omega\rangle are projective systems of groups of cardinality less than \lambda and suppose that for every n< \omega there is a homorphism \sigma:H_n\rightarrow G_n such that all the diagrams commute. If for every \mu< \lambda there exists \langle f_i\in G_{\omega} \;:\;i< \mu\rangle such that i\neq j\Longrightarrow f_if_j^{-1}\not\in\sigma_{\omega}(H_{\omega}) then there exists \langle f_i\in G_{\omega} \;:\;i < 2^{\lambda}\rangle such that i\neq j\Longrightarrow f_if_j^{-1}\not\in\sigma_{\omega}(H_{\omega}).
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@article{Sh:302a,
 author = {Grossberg, Rami P. and Shelah, Saharon},
 title = {{On cardinalities in quotients of inverse limits of groups}},
 journal = {Math. Japon.},
 fjournal = {Mathematica Japonica},
 volume = {47},
 number = {2},
 year = {1998},
 pages = {189--197},
 issn = {0025-5513},
 mrnumber = {1615081},
 mrclass = {20E18},
 note = {\href{https://arxiv.org/abs/math/9911225}{arXiv: math/9911225}},
 arxiv_number = {math/9911225}
}