关于有限群的三重分解
On triple factorisations of finite groups
摘要：
本文引入并发展了一个研究有限群$G$的$G=ABA$形式的三重分解的一般框架，其中包含$G$的$A$和$B$子群.我们称这样的分解为非退化的，如果$G\neq AB$。考虑$G$的行动由右乘$B$的右陪集导致一个非平凡的上限为$|G|$通过应用有关子集的限制运动的结果。对于$A<C<G$和$B<D<G$，即使$G=ABA$是非简并的，因子分解$G=CDC$也可能是简并的。类似地，形成双因子可能导致退化的三因子分解。一个理由是减少非退化的三重因式分解的研究，其中$G$的行为忠实和recittively对陪集的$A$。这涉及到三重因式分解的圈积构造的研究。
英文摘要：
This paper introduces and develops a general framework for studying triple factorisations of the form $G=ABA$ of finite groups $G$, with $A$ and $B$ subgroups of $G$. We call such a factorisation nondegenerate if $G\neq AB$. Consideration of the action of $G$ by right multiplication on the right cosets of $B$ leads to a nontrivial upper bound for $|G|$ by applying results about subsets of restricted movement. For $A<C<G$ and $B<D<G$ the factorisation $G=CDC$ may be degenerate even if $G=ABA$ is nondegenerate. Similarly forming quotients may lead to degenerate triple factorisations. A rationale is given for reducing the study of nondegenerate triple factorisations to those in which $G$ acts faithfully and primitively on the cosets of $A$. This involves study of a wreath product construction for triple factorisations.
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