Sylowizers in locally finite groups
Sylowizers in locally finite groups
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DOI：
10.1090/s0002-9939-1974-0349848-7
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发表时间：
1974-02
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影响因子：
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通讯作者：
M. J. Tomkinson
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文献类型：
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作者：
M. J. Tomkinson
W. Gaschi7tz 对于有限可溶群的结果被扩展到两类局部有限、局部可溶群。 Sylowizers 的概念由 W. GaschiYtz 提出[2]。如果 R 是群 G 的 v 子群，则 G 中 R 的 7-Sylowizer 是关于包含 R 作为 Sylow u 子群的 G 最大的子群 S。 [7T 表示一组素数，Sylow 7..subgroup 只是一个最大 77 子群。] 简单的 Zorn 引理论证表明任何 77 子群 R 的 uT-Sylowizers 必须始终存在。 Gaschdtz 证明了以下共轭定理：设 G 是有限可溶群，R 是 G 的某个 Sylow u 子群 P 的正规子群。然后 G 中 R 的 u-Sylowizer 在 G 中共轭。本文的目的是将此结果扩展到周期性局部可溶 FC 群的类 3 和 U 类，定义如下： G E U 当且仅当 G 是局部有限的并且对于每个 H < G 且对于每个素数集u7，H 的 Sylow ur 子群在 H 中共轭。[关于 ( 和 U 的必要结果可以分别在[5]和[1]中找到。]有限群的证明涉及通常考虑最小阶反例。这种方法不能用于无限群，我们对 R 的 77-Sylowizers 进行了更直接的构造，尽管这种构造是基于 Gaschiutz 证明中使用的思想。尽管我们只证明了定理的扩展如图 7 和 U 所示，u-Sylowizers 的构造是在更广泛的群类中进行的，我们将 X 定义为上 u-可分局部有限群 G 的类，使得只要 K '1 H < G 且 P 是 H 的 Sylow u7-子群，PK/K 就是 H/K 的 Sylow u7-子群。 4-群显然是上 u-可分的，并且其 Sylow 77-子群也具有必要的同态性质。 [5, 4.1 (iii)]。编辑们于 1973 年 4 月 23 日收到 U 组，并于 1973 年 8 月 7 日修订。AMS (MOS) 主题分类 (1970)。
A result of W. Gaschi7tz for finite soluble groups is extended to two classes of locally finite, locally soluble groups. The concept of Sylowizers has been introduced by W. GaschiYtz [2]. If R is a v-subgroup of the group G, then a 7-Sylowizer of R in G is a subgroup S of G maximal with respect to containing R as a Sylow u-subgroup. [7T denotes a set of primes and a Sylow 7..subgroup is simply a maximal 77subgroup.] A straightforward Zorn's lemma argument shows that uT-Sylowizers of any 77-subgroup R must always exist. Gaschdtz proved the following conjugacy theorem: Let G be a finite soluble group and R a normal subgroup of some Sylow u-subgroup P of G. Then the u-Sylowizers of R in G are conjugate in G. It is our aim in this note to extend this result to the class 3 of periodic locally soluble FC-groups and the class U defined by: G E U if and only if G is locally finite and for each H < G and for each set of primes u7, the Sylow ur-subgroups of H are conjugate in H. [The necessary results about ( and U may be found in [5] and [1] respectively.] The proof for finite groups involves the usual consideration of a counterexample of minimal order. This method cannot be employed for infinite groups and we make a more direct construction of the 77-Sylowizers of R, although this construction is based on the ideas used in Gaschiutz's proof. Although we only prove the extension of the theorem for 7 and U, the construction of the u-Sylowizers is carried out in a much wider class of groups. We define X to be the class of upper u-separable locally finite groups G such that PK/K is a Sylow u7-subgroup of H/K whenever K '1 H < G and P is a Sylow u-subgroup of H. A 4-group is clearly upper u-separable and also its Sylow 77-subgroups have the necessary homomorphism property [5, 4.1 (iii)]. Also U-groups are Received by the editors April 23, 1973 and, in revised form, August 7, 1973. AMS (MOS) subject elassifications (1970). Primary 20E25.