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The cofinality of normal subgroups and homomorphic images of infinite symmetric groups. (English) Zbl 1096.20003

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 \(\alpha\leq k\), \(\text{Sym}_\alpha(k)=\{\pi\in\text{Sym}(k):|\text{supp}(\pi )|<\alpha\}\).

MSC:

20B30 Symmetric groups
20B35 Subgroups of symmetric groups
03E04 Ordered sets and their cofinalities; pcf theory
03E75 Applications of set theory
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