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- W2015615060 abstract "Throughout this paper κ denotes an infinite cardinal, S = Sym( κ ) and G is a subgroup of S . We shall be seeking the subgroups G with [ S : G ] < 2 κ . In [2], the following result was proved. Theorem 1. If [ S: G ] ≤ κ then there exists a subset Δ of k such that ∣Δ∣ < k and S (Δ) ≤ G . Here S (Δ) = Sym( K/⊿ ) is the pointwise stabilizer of Δ in S . However, the converse of Theorem 1 is not true. For if cf( κ ) ≤ ∣Δ∣ < κ , then [ S : S (Δ) ] ≥ κ cf( κ ) > κ . This suggests that a substantially sharpened version of Theorem 1 may be true. Question 1 [2]. Is it provable in ZFC, or even in ZFC with GCH, that if [ S : G ] ≤ κ then there is a subset Δ of κ such that ∣Δ∣ < cf( κ ) and S (Δ) ≤ G ? At least two of the authors of [2] made a serious attempt to answer the above question positively. In §3, we shall see that they were essentially trying to prove that measurable cardinals do not exist. The following result, due independently to Semmes [5] and Neumann [2], suggests a second way in which Theorem 1 might be improved. Theorem 2. If k = ℵ 0 and then there is a finite subset Δ of k such that S (Δ) ≤ G . Question 2 [2]. Is it provable in ZFC that if [ S : G ] < 2 κ then there is a subset Δ of κ such that ∣Δ∣ < κ and S (Δ) < G ? This question will be answered negatively in §4." @default.
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- W2015615060 date "1989-03-01" @default.
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- W2015615060 title "Subgroups of small index in infinite symmetric groups. II " @default.
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- W2015615060 doi "https://doi.org/10.2307/2275018" @default.
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