Fischer, Vera Maximal cofinitary groups revisited. (English) Zbl 1372.03096 Math. Log. Q. 61, No. 4-5, 367-379 (2015). Summary: Let \(\kappa\) be an arbitrary regular infinite cardinal and let \(\mathcal C\) denote the set of \(\kappa\)-maximal cofinitary groups. We show that if \(\mathsf {GCH}\) holds and \(\mathcal C\) is a closed set of cardinals such that 1. \(\kappa + \in C\), \(\forall\nu\in C(\nu \geq \kappa^ + )\),2. if \(|C| \geq \kappa ^+\) then \([\kappa^ + , |C|] \subseteq C\),3.\( \forall \nu \in C(\mathrm{cof}(\nu) \leq \kappa\to \nu ^+ \in C)\), then there is a generic extension in which cofinalities have not been changed and such that \(C = \{|\mathcal G| :\mathcal G \in \mathcal C\}\). The theorem generalizes a result of J. Brendle et al. [J. Algebra 232, No. 1, 209–225 (2000; Zbl 0966.20001)] regarding the possible sizes of maximal cofinitary groups.Our techniques easily modify to provide analogous results for the spectra of maximal \(\kappa\)-almost disjoint families in \([\kappa]^ \kappa \), maximal families of \(\kappa\)-almost disjoint permutations on \(\kappa\) and maximal families of \(\kappa\)-almost disjoint functions in \({}^\kappa\kappa\). In addition we construct a \(\kappa\)-Cohen indestructible \(\kappa\)-maximal cofinitary group and so establish the consistency of \(\mathfrak a _g (\kappa) < \mathfrak d(\kappa)\), which for \(\kappa =\omega\) is due to Y. Zhang [Arch. Math. Logic 39, No. 1, 41–52 (2000; Zbl 0954.20001)]. Cited in 6 Documents MSC: 03E35 Consistency and independence results 20B07 General theory for infinite permutation groups 03F05 Cut-elimination and normal-form theorems 03E17 Cardinal characteristics of the continuum 20A15 Applications of logic to group theory Keywords:\(\kappa\)-maximal cofinitary groups; GCH; cofinalities; spectra; maximal \(\kappa\)-almost disjoint families Citations:Zbl 0966.20001; Zbl 0954.20001 PDFBibTeX XMLCite \textit{V. Fischer}, Math. Log. Q. 61, No. 4--5, 367--379 (2015; Zbl 1372.03096) Full Text: DOI References: [1] Blass, Israel Mathematical Conference Proceedings Vol. 6 pp 63– (1993) [2] Brendle, Contemporary Mathematics Vol. 302 pp 1– (2002) [3] Brendle, The almost-disjointness number may have countable cofinality, Trans. Am. Math. Soc. 355 (7) pp 2633– (2003) · Zbl 1061.03051 [4] Brendle, Uniformity of the meager ideal and maximal cofinitary groups, J. Algebra 232 (1) pp 209– (2000) · Zbl 0966.20001 [5] Fischer, Template iterations and maximal cofinitary groups, Fundam. Math. 230 (3) pp 205– (2015) · Zbl 1373.03089 [6] S. D. Friedman Invariant Descriptive Set Theory, unpublished lecture notes 2013 [7] Hechler, Short complete nested sequences in {\(\beta\)}NN and small maximal almost-disjoint families, Gen. Topology Appl. 2 pp 139– (1972) · Zbl 0246.02047 [8] K. Kunen Set theory. An introduction to independence proofs Studies in Logic and the Foundations of Mathematics Vol. 102 North-Holland 1983 [9] S. Shelah O. Spinas MAD spectra, J. Symb. Log., to appear · Zbl 1367.03093 [10] Zhang, Maximal cofinitary groups, Arch. Math. Log. 39 (1) pp 41– (2000) · Zbl 0954.20001 [11] Zhang, Permutation groups and covering properties, J. Lond. Math. Soc. (2) 63 (1) pp 1– (2001) · Zbl 1018.20001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.