论文标题
在具有和弦功率图的组上，包括有限简单组的分类
On groups with chordal power graph, including a classification in the case of finite simple groups
论文作者
论文摘要
我们证明了功率图是弦的组的结构上的各种属性。具有该特性的Nilpotent群体已由Manna，Cameron和Mehatari [Electronic of Combinatorics，2021年]进行了分类。在这里，我们将有限的简单基团与弦元电源图分类，相对于典型的数字理论oracles。我们这样做是通过为有限基团的功率图中的长周期的存在和不存在的几个足够条件设计。我们检查了其他天然组类别，包括特殊的线性，对称，广义二面体和Quathnion组，我们用弦式功率图表征了直接产品。从而将分类问题缩小为直接不可分解的组，我们进一步获得了可能的SOCLES列表。最后，我们在弦弦功率图中的诱发路径的长度上给出了一般界限，这为将分类从简单组中推向了另一条潜在的道路。
We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by Manna, Cameron and Mehatari [The Electronic Journal of Combinatorics, 2021]. Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic oracles. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.
