论文标题
扰乱产生的组
Groups generated by derangements
论文作者
论文摘要
我们检查了$ g $ dyangements产生的及时排列组$ g $的子组$ d（g）$。我们的主要结果绑定了该子组的索引：我们认为，如果$ g $具有$ n $，而不是frobenius组，则$ | g：d（g）| \ leqslant \ sqrt \ sqrt {n} -1 $;我们证明了这一点，除非$ g $是一个原始的仿射组。对于仿制组，我们将猜想转化为$ | h：r（h）| $的同等形式，其中$ h $是有限矢量空间上的线性群体，而$ r（h）$是由具有特征值〜$ 1 $的元素产生的$ h $的亚组。 如果$ g $是Frobenius组，则$ d（g）$是Frobenius内核，因此$ g/d（g）$是Frobenius补充的同构。我们举了一些示例，其中$ d（g）\ ne g $，并检查$ g/d（g）$的群体理论结构；特别是，我们构建了$ g/d（g）$的$ g $，不是Frobenius的补充。
We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group, then $|G:D(G)|\leqslant\sqrt{n}-1$; we prove this except when $G$ is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding $|H:R(H)|$, where $H$ is a linear group on a finite vector space and $R(H)$ is the subgroup of $H$ generated by elements having eigenvalue~$1$. If $G$ is a Frobenius group, then $D(G)$ is the Frobenius kernel, and so $G/D(G)$ is isomorphic to a Frobenius complement. We give some examples where $D(G)\ne G$, and examine the group-theoretic structure of $G/D(G)$; in particular, we construct groups $G$ in which $G/D(G)$ is not a Frobenius complement.
