Subgroups of small index in infinite symmetric groups. II
Subgroups of small index in infinite symmetric groups. II
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无限对称群中的小指数子群。
DOI：
10.2307/2275018
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发表时间：
1986
影响因子：
0.6
通讯作者：
S. Thomas
中科院分区：
文献类型：
--
作者：
S. Shelah;S. Thomas
在本文中，κ表示一个无限基数，S = Sym(κ)， G是S的一个子群。我们将寻求[S: G] < 2κ的子群G。在[2]中，证明了以下结果。定理1。若[S: G]≤κ，则存在k的一个子集Δ，使得∣Δ∣< k且S（Δ）≤G。这里S（Δ） = Sym（k /⊿）是S中Δ的点向稳定器，但定理1的逆命题不成立。如果cf(κ)≤∣Δ∣<κ,那么(S, S(Δ))≥κcf(κ)>κ。这表明定理1的一个本质上强化的版本可能是正确的。问题1 b[2]。是否可以证明在ZFC中，或者甚至在GCH的ZFC中，如果[S: G]≤κ，那么存在一个κ的子集Δ使得∣Δ∣< cf（κ）和S（Δ）≤G？b[2]的作者中至少有两位认真尝试正面回答上述问题。在§3中，我们将看到，他们实质上是在试图证明可测的基数不存在。下面的结果，独立于Semmes[5]和Neumann[2]，提出了定理1可能改进的第二种方法。定理2。如果k = 0，则存在k的一个有限子集Δ，使得（Δ）≤g。在ZFC中是否可以证明，如果[S: G] < 2κ，那么存在一个κ的子集Δ使得∣Δ∣< κ且S（Δ） < G？这个问题将在§4中得到否定的回答。
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] < 2κ. In [2], the following result was proved. Theorem 1. If [S: G] ≤ κthen 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)] ≥ κcf(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 = ℵ0andthen there is a finite subset Δ of k such thatS(Δ) ≤ G. Question 2 [2]. Is it provable in ZFC that if [S: G] < 2κ then there is a subset Δ of κ such that ∣Δ∣ < κ and S(Δ) < G? This question will be answered negatively in §4.