三重积性质三元组的搜索和测试算法
Search and test algorithms for Triple Product Property triples
摘要：
2003年，COHN和UMANS提出了一种快速矩阵乘法的群论方法。这涉及找到满足三重积性质（TPP）的群$G$的大子集作为约束矩阵乘法的指数$ω$的手段。我们提出了两个新的特征的TPP，这是有用的理论考虑和TPP测试算法。有了这个，我们描述了所有已知的TPP测试，并在GAP算法中实现它们。我们还比较了它们的运行时间。此外，我们表明，搜索子群TPP三元组的非平凡大小的非阿贝尔群可以限制到该组的所有非正规子群的集合。最后，我们描述了极大子群和子集TPP三元组的蛮力搜索算法。此外，我们提出的结果子集蛮力搜索的顺序小于25和选定的结果的子群蛮力搜索2组，$SL(n,q)$和$PSL(2,q)$的所有群体。
英文摘要：
In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $ω$ of matrix multiplication. We present two new characterizations of the TPP, which are useful for theoretical considerations and for TPP test algorithms. With this we describe all known TPP tests and implement them in GAP algorithms. We also compare their runtime. Furthermore we show that the search for subgroup TPP triples of nontrivial size in a nonabelian group can be restricted to the set of all nonnormal subgroups of that group. Finally we describe brute-force search algorithms for maximal subgroup and subset TPP triples. In addition we present the results of the subset brute-force search for all groups of order less than 25 and selected results of the subgroup brute-force search for 2-groups, $SL(n,q)$ and $PSL(2,q)$.
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