关于多循环猜想
On the polycirculant conjecture
摘要：
本文发展了$k$轨道理论的基础。该理论为研究群和多维对称性开辟了一条新的简单途径。研究了$k$ -轨道的组合对称性与其自同构群之间的关系，以及$k$ -轨道的局部性.发现了$m>2$的2-闭群与$m$ -闭群之间的差异.说明了Petersen图自同构群$n$ -轨道的特殊性质。证明了任何非平凡的本原群都包含一个传递的非本原子群，从而证明了点传递图（2-闭群）的自同构群包含一个正则元（多循环猜想）.利用$k$ -轨道理论的方法，考虑了有限群的置换表示的不同可能性，并证明了相对于有限群的结构的描述，信息量最大的是最低次的置换表示.利用这个表示，得到了W. Feit，J.G.汤普森定理：奇阶群的可解性。给出了有限群的最低次表示的足够简单的结构，并找到了构造有限群的简单全不变量的方法。最后，利用$k$ -轨道理论的方法，得到了图同构问题的一个可能的多项式解。
英文摘要：
In the paper the foundation of the $k$-orbit theory is developed. The theory opens a new simple way to the investigation of groups and multidimensional symmetries. The relations between combinatorial symmetry properties of a $k$-orbit and its automorphism group are found. It is found the local property of a $k$-orbit. The difference between 2-closed group and $m$-closed group for $m>2$ is discovered. It is explained the specific property of Petersen graph automorphism group $n$-orbit. It is shown that any non-trivial primitive group contains a transitive imprimitive subgroup and as a result it is proved that the automorphism group of a vertex transitive graph (2-closed group) contains a regular element (polycirculant conjecture). Using methods of the $k$-orbit theory, it is considered different possibilities of permutation representation of a finite group and shown that the most informative, relative to describing of the structure of a finite group, is the permutation representation of the lowest degree. Using this representation it is obtained a simple proof of the W. Feit, J.G. Thompson theorem: Solvability of groups of odd order. It is described the enough simple structure of lowest degree representation of finite groups and found a way to constructing of the simple full invariant of a finite group. To the end, using methods of $k$-orbit theory, it is obtained one of possible polynomial solutions of the graph isomorphism problem.
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