arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2004.01950math.GR
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乱序生成的群

Groups generated by derangements

R. A. Bailey, Peter J. Cameron, Michael Giudici, Gordon F. Royle

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摘要：

研究了由$G$中的错位生成的可迁置换群$G$的子群$D(G)$.我们的主要结果约束指数的子群：我们猜想，如果$G$度$n$和不是Frobenius群，那么$|G:D(G)|\leqslant\sqrt{n}-1$ ;我们证明了这一点，除了当$G$是一个原始仿射群。对于仿射群，我们将猜想转化为关于$|H:R(H)|$的等价形式，其中$H$是有限向量空间上的线性群，$R(H)$是由具有特征值~ $1$的元素生成的$H$的子群。如果$G$是Frobenius群，则$D(G)$是Frobenius核，因此$G/D(G)$同构于Frobenius补。我们给出了一些例子，其中$D(G)\ne G$，并研究了$G/D(G)$的群论结构，特别是，我们构建的群体$G$中，$G/D(G)$是不是一个Frobenius补。

英文摘要：

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.

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