有限置换群的第二类商

Pub Date : 2023-06-29 DOI:10.1515/jgth-2022-0214

H. Meng, Xiuyun Guo

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引用次数: 0

摘要

设𝐺是有限集合上的一个置换群，设𝑝是一个素数。本文证明了𝐺的最大二类𝑝-quotient的阶数最多为p n/p p^{n/p}(如果p=2 p=2，则为2 3 × n/4 × 2^{3n/4})，其中𝑛为𝐺的一个Sylow𝑝-subgroup移动的点数。进一步，我们描述了最大的二类𝑝-quotients可以达到这样一个界限的群。这延伸了Kovács和Praeger从1989年开始的早期工作。

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Class-two quotients of finite permutation groups

Abstract Let 𝐺 be a permutation group on a finite set and let 𝑝 be a prime. In this paper, we prove that the largest class-two 𝑝-quotient of 𝐺 has order at most p n / p p^{n/p} (or 2 3 ⁢ n / 4 2^{3n/4} if p = 2 p=2 ), where 𝑛 is the number of points moved by a Sylow 𝑝-subgroup of 𝐺. Further, we describe the groups whose largest class-two 𝑝-quotients can reach such a bound. This extends earlier work of Kovács and Praeger from 1989.

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