论文标题
置于稳定器的置换组
Permutation groups with restricted stabilizers
论文作者
论文摘要
修复一个正整数$ d $，让$γ_d$是与交替组$ a_d $同构的有限组类别。 Babai，Cameron和Pálfy在1980年代研究了$γ_D$中的这些组，他们确定了与该特性的原始置换组的顺序，这些属性发现了广泛的应用。随后，还获得了此类组的基础大小的结果。在本文中，我们通过限制其点稳定剂来代替该组的结构条件，并得出相似的结论，有时甚至更强。例如，我们证明有一个线性函数$ f $，以便任何有限原始组的基本大小，其中$γ_d$中的点稳定器最多为$ f（d）$。这概括了第一作者对具有可解决点稳定器的原始群体的最新结果。对于非种植原始群体，我们获得更强的结果，假设$ C $点的稳定器位于$γ_D$中。我们还表明，如果$ g $是$ c $ coppoint稳定器的任何置换$ n $的排列组，则为$γ_d$，则$ | g | \ leqslant（（（1+O_C（1））D/E）^{n-1} $。该渐近地扩展并改善了Babai，Cameron和Pálfy在γ_D$中获得$ g \ $ g \ $ | g | $上的$ d^{n-1} $上限。
Fix a positive integer $d$ and let $Γ_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $Γ_d$ were studied by Babai, Cameron and Pálfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function $f$ such that the base size of any finite primitive group with point stabilizers in $Γ_d$ is at most $f(d)$. This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of $c$ points lie in $Γ_d$. We also show that if $G$ is any permutation group of degree $n$ whose $c$-point stabilizers lie in $Γ_d$, then $|G| \leqslant ((1+o_c(1))d/e)^{n-1}$. This asymptotically extends and improves a $d^{n-1}$ upper bound on $|G|$ obtained by Babai, Cameron and Pálfy assuming $G \in Γ_d$.
