# Sh:524

• Shelah, S., & Thomas, S. (1997). The cofinality spectrum of the infinite symmetric group. J. Symbolic Logic, 62(3), 902–916.
• Abstract:
A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The cofinality spectrum of G, written CF(S), is the set of regular cardinals \lambda such that G can be expressed as the union of a chain of \lambda proper subgroups. The cofinality of G, written c(G), is the least element of CF(G). We show that it is consistent that CF(S) is quite a bizarre set of cardinals. For example, we prove

Theorem (A): Let T be any subset of \omega\setminus \{0\}. Then it is consistent that \aleph_n \in CF(S) if and only if n\in T.

One might suspect that it is consistent that CF(S) is an arbitrarily prescribed set of regular uncountable cardinals, subject only to the above mentioned constraint. This is not the case.

Theorem (B): If \aleph_n \in CF(S) for all n\in \omega\setminus \{0\}, then \aleph_{\omega+1} \in CF(S).

• Version 1994-12-26_10 (18p) published version (16p)
Bib entry
@article{Sh:524,
author = {Shelah, Saharon and Thomas, Simon},
title = {{The cofinality spectrum of the infinite symmetric group}},
journal = {J. Symbolic Logic},
fjournal = {The Journal of Symbolic Logic},
volume = {62},
number = {3},
year = {1997},
pages = {902--916},
issn = {0022-4812},
mrnumber = {1472129},
mrclass = {03E35 (20B30)},
doi = {10.2307/2275578},
note = {\href{https://arxiv.org/abs/math/9412230}{arXiv: math/9412230}},
arxiv_number = {math/9412230}
}