Groupwise density and the cofinality of the infinite symmetric group (Q1283121)

For a group \(G\) which is not finitely generated, the cofinality of \(G\), written \(c(G)\), is defined to be the least cardinal \(\lambda\) such that \(G\) is the union of a chain of \(\lambda\) proper subgroups. For an infinite cardinal \(\kappa\), denote \(c({\mathfrak{Sym}}(\kappa))\) by \(c_\kappa\). \textit{H.~D. Macpherson} and \textit{P.~M. Neumann} [J. Lond. Math. Soc. (2) 42, 64-84 (1990; Zbl 0668.20005)] proved that \(c_\kappa >\kappa\), for any \(\kappa\). In particular, \(\omega <c_\omega\leq 2^\omega\); so \(c_\omega = 2^\omega\) if the Continuum Hypothesis is true. \textit{J.~D. Sharp} and the author [Notre Dame J. Formal Logic 35, 328-345 (1994; Zbl 0824.03027)] showed that Martin's Axiom implies \(c_\omega = 2^\omega\). In Arch. Math. Logic 34, 33-45 (1995; Zbl 0818.03027) they studied the relationship between \(c_\omega\) and two known invariants of the continuum, the dominating number \(\mathfrak d\) and the bounding number \(\mathfrak b\). In the paper under review the author considers the relationship between \(c_\omega\) and some other invariants of the continuum, \(\mathfrak u\) and \(\mathfrak g\). Here \(\mathfrak u\) is the smallest cardinality of an ultrafilter base,and \(\mathfrak g\) is the groupwise density number which is defined as follows. A family \(\mathcal G\) of infinite subsets of \(\omega\) is called groupwise dense if, first, whenever \(X\in\mathcal G\) then \(Y\in\mathcal G\) for any infinite \(Y\subseteq\omega\) with finite \(Y\smallsetminus X\), and, second, for any partition \(\{I_n: n\in\omega\}\) of \(\omega\) into finite intervals there exists an infinite \(J\subseteq\omega\) such that \(\bigcup\{I_n: n\in J\}\) is in \(\mathcal G\). The groupwise density number \(\mathfrak g\) is defined to be the least \(\lambda\) such that there exists a collection \(\{{\mathcal G}_\alpha : \alpha<\lambda\}\) of groupwise dense families with empty \(\bigcap\{{\mathcal G}_\alpha : \alpha<\lambda\}\). The author proves the consistency of \(\omega_1={\mathfrak u}<c_\omega=\omega_2=2^\omega\) and observes that one of the results of Sharp and himself easily implies the consistency of \(\omega_1=c_\omega<{\mathfrak u}=2^\omega\). He discusses the open conjecture that both \(c_\omega <\mathfrak g\) and \(c_\omega >\mathfrak g\) are consistent with ZFC, and shows that \(c_\omega=\omega_1\) in one of the models with \({\mathfrak g}=\omega_1<2^\omega\) constructed by A.~Blass and S.~Shelah.