Infinite permutation groups II. Subgroups of small index
Infinite permutation groups II. Subgroups of small index
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无限排列群 II.
DOI：
10.1016/0021-8693(89)90212-3
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发表时间：
1989
影响因子：
0.9
通讯作者：
J.K Truss
中科院分区：
文献类型：
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作者：
J.K Truss
狄克逊、诺依曼和托马斯证明了：如果52是可数无限集，H是Q上指数小于2no的全对称群Sym（SZ）的子群，则H包含Sym（Q）中52的有限子集的点态稳定子。对于任意群G，（它通常是Sym（Q）的一个子群，对于一些52，ISZJ= N），作为“关于G的小指数子群的猜想”，Evans [S]证明了该猜想对维数为K的向量空间的一般线性群成立，在可数Bruyns [Z]在G= Aut Q（Q到自身的同胚群）的情形下得到了关于猜想的部分信息。这个猜想在[3]中针对两种特别有趣的情况，Aut Q和A（Q）（后者是Q的保序置换群）进行了阐述。Macpherson [S]更有野心，并建议该猜想应该适用于任何NO-范畴结构的自同构群（尽管他将假设IG：HI<2 'O削弱为IG：HI< K，），Evans [6]展示了如何使用G的“闭”子群的思想重新表述该猜想。最近Hrushovski发现了一个反例一般情况下的猜想，虽然它似乎仍然可能，它将举行广泛的一类实例。这里的目的是建立在[3]中提出的两个主要情况下的猜想的真理。我们也将在其他一些情况下证明这一点，主要是可数无原子布尔代数的自同构群。这与Aut C同构为graq，其中C是康托不连续统，尽管不是置换群（实际上Aut C的度数为2“”）。因此，虽然我们主要使用Aut C（或者更确切地说Aut 2”），但在这一组中表达结果时，结果的陈述会有相当大的不同。也就是说：如果H是Aut C的指数小于2X 0 494的子群
It was shown by Dixon, Neumann, and Thomas [3] that if 52 is a countably infinite set, and H is a subgroup of the full symmetric group Sym (SZ) on Q of index less than 2no, then H contains the pointwise stabilizer in Sym (Q) of a finite subset of 52. Referring to the analogous statement for an arbitrary group G (which will usually be a subgroup of Sym (Q) for some 52 with ISZJ= N,) as the “conjecture on subgroups of small index for G” it was shown by Evans [S] that the conjecture holds for the general linear group of a vector space of dimension K, over a countable (or finite) division ring, and Bruyns [Z] obtained partial information about the conjecture in the case where G= Aut Q, the group of homeomorphisms of Q to itself. The conjecture was formulated in [3] for two particularly interesting cases, Aut Q and A (Q)(the latter being the group of order-preserving permutations of Q). Macpherson [S] was more ambitious and suggested that the conjecture should hold for the automorphism group of any NO-categorical structure (though he weakened the hypothesis IG: HI< 2’O to IG: HI< K,), and Evans [6] showed how to reformulate the conjecture using ideas about “closed” subgroups of G. Recently Hrushovski has found a counter-example to the general case of the conjecture, though it still seems likely that it will hold in a wide class of instances. The object here is to establish the truth of the conjecture in the two main cases raised in [3]. We shall prove it in some other cases too, principally the group of automorphisms of the countable atomless Boolean algebra. This is isomorphic as a graq to Aut C, where C is the Cantor discontinuum, though not as a permutation group (in fact Aut C has degree 2’“). Thus although we work mainly with Aut C (or rather Aut 2”) the statement of the result comes out rather differently when expressed in this group. Namely it says: if H is a subgroup of Aut C of index less than 2X0 494