论文标题
Moufang循环的功率图
Power graphs of Moufang loops
论文作者
论文摘要
两组和半群的功率图已经广泛研究。虽然可以将准群的功率图类似于组的幂图，但迄今为止，对准群和环的功率图很少进行。在本文中，我们通过解决彼得·卡梅伦（Peter Cameron）提出的问题来开始将组的功率图上的结果传输到循环的上下文：如果两个moufang循环具有同构无向功率图，那么它们是否必须具有同构的定向功率图？众所周知，具有同构无向功率图的两个组必须具有同构的定向功率图。我们能够通过显示具有同构无向功率图具有同构有向功率图的Moufang循环将结果扩展到Moufang循环。在此过程中，我们研究了一类Moufang循环，我们将其称为广义八循环，并且与广义的季节组相似。
Power graphs of both groups and semigroups have been widely studied. While the power graph of a quasigroup can be defined analogously to that of a group, power graphs of quasigroups and loops have thus far been little studied. In this paper we begin transferring results on the power graphs of groups to the context of loops by addressing a question posed by Peter Cameron: if two Moufang loops have isomorphic undirected power graphs, must they have isomorphic directed power graphs? It is known that two groups with isomorphic undirected power graphs must have isomorphic directed power graphs. We are able to extend that result to Moufang loops by showing that Moufang loops with isomorphic undirected power graphs have isomorphic directed power graphs. Along the way we investigate a class of Moufang loops which we term the generalized octonion loops and behave similarly to the generalized quaternion groups.
