不可数一般线性群的小指数特征

THE SMALL INDEX PROPERTY OF UNCOUNTABLE GENERAL LINEAR GROUPS

使用连续统假设，证明了如下结论：设Ｖ是可数数域Ｆ上的－维向量空间，Ｇ＝ＧＬ（Ｖ）是Ｖ的一般线性群，Ｈ是Ｇ的子群，并且｜Ｇ：Ｈ｜＜２１，则存在Ｖ的可数维子空间Ｘ，使得Ｇ（Ｘ）≤Ｈ≤Ｇ｛ｘ｝。

Using CH, the author shows the following result. Let V be a vector space of dimension over a countable field and let G=GL(V) be the group of all invertible linear transformations on V. Let H be a subgroup of G of index less than 2. Then there exists a countable-dimensional subspace X of V such that G(x)≤H≤G {x}.