Felgner, Ulrich; Schatz, Torsten The cofinality of normal subgroups and homomorphic images of infinite symmetric groups. (English) Zbl 1096.20003 Commun. Algebra 33, No. 8, 2601-2606 (2005). The cofinality \(c(G)\) of a group \(G\) (which is not finitely generated) is defined to be the least cardinal \(\mu\) such that \(G\) can be expressed as the union of an increasing chain of \(\mu\) proper subgroups. If \(k\) is an infinite cardinal, then \(\text{Sym}(k)\) (\(\text{Alt}(k)\)) denotes the group of all permutations (alternating) of the set \(k=\{\alpha:\alpha<k\}\). The aim of this paper is to study the cofinality of the proper normal subgroups of \(\text{Sym}(k)\) and the homomorphic images of \(\text{Sym}(k)\) and its normal subgroups. So, the authors prove that: (i) If \(k\) and \(\alpha\) are infinite cardinals with \(\alpha\leq k\), then \(c(\text{Sym}(k)/\text{Sym}_\alpha(k))=c(\text{Sym}(k)/\text{Alt}(k))=c(\text{Sym}(k))\); (ii) If \(k,\alpha\) and \(\theta\) are infinite cardinals with \(\alpha<\theta\leq k\), then \(c(\text{Sym}_\theta(k)/\text{Sym}_\alpha(k))=c(\text{Sym}_\theta(k)/\text{Alt}(k))=\text{cf}(\theta)\), where if \(\alpha\) is an infinite cardinal with \(\alpha\leq k\), \(\text{Sym}_\alpha(k)=\{\pi\in\text{Sym}(k):|\text{supp}(\pi )|<\alpha\}\). Reviewer: Dimitru Buşneag (Craiova) MSC: 20B30 Symmetric groups 20B35 Subgroups of symmetric groups 03E04 Ordered sets and their cofinalities; pcf theory 03E75 Applications of set theory Keywords:cofinalities; infinite symmetric groups; alternating groups PDFBibTeX XMLCite \textit{U. Felgner} and \textit{T. Schatz}, Commun. Algebra 33, No. 8, 2601--2606 (2005; Zbl 1096.20003) Full Text: DOI References: [1] Felgner U., Forum Mathematicum 5 pp 505– (1993) · Zbl 0791.20003 · doi:10.1515/form.1993.5.505 [2] Macpherson H. D., J. London Math. Soc. 42 (2) pp 64– (1990) · Zbl 0668.20005 · doi:10.1112/jlms/s2-42.1.64 [3] Scott W. R., Group Theory. (1964) [4] Sharp J. D., J. Symbolic Logic 60 pp 892– (1995) · Zbl 0853.03014 · doi:10.2307/2275763 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.