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arXiv 2403.08129math.GR
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有限群的Solvabilizer数

Solvabilizer Numbers of Finite Groups

Banafsheh Akbari, Tuval Foguel, Jack Schmidt

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AI总结：

本文研究非可解有限群的solvabilizer数，通过定义solvabilizer覆盖并求其最小基数，揭示可解性与群结构之间的内在联系。

摘要：

考虑一个非可解有限群G，其中R(G)表示G的可解根。对于G中的任意元素x，x在G中的solvabilizer，记作Sol_G(x)，定义为G中所有使得由x和y生成的子群可解的元素y的集合。值得注意的是，整个G可以表示为所有x属于G\R(G)的相应solvabilizer的并集：$G = \cup_{x\in G\R(G)} Sol_G(x)$。G的一个solvabilizer覆盖由G\R(G)的一个子集X刻画，使得G= \cup_{x\in X} Sol_G(x)。G的solvabilizer数则定义为G的所有solvabilizer覆盖的最小基数。本文深入探讨了多种非可解有限群G的solvabilizer数，揭示了可解性与这些群的结构之间的相互作用。

英文摘要：

Consider a nonsolvable finite group G, where R(G) represents the solvable radical of G. For any element x in G, the solvabilizer of x in G, denoted by Sol_G(x), is defined as the set of all elements y in G such that the subgroup generated by x and y is solvable. Notably, the entirety of G can be expressed as the union over all x in G\R(G) of their respective solvabilizers: $G = \cup_{x\in G\R(G)} Sol_G(x). A solvabilizer covering of G is characterized by a subset X of G\R(G) such that G= \cup_{x\in X} Sol_G(x). The solvabilizer number of G is then defined as the minimum cardinality among all solvabilizer coverings of G. This paper delves into the exploration of the solvabilizer number for diverse nonsolvable finite groups G, shedding light on the interplay between solvability and the structure of these groups.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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