Search and test algorithms for Triple Product Property triples

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 $\omega$ 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 $PSL(2,q)$.

💡 Research Summary

The paper investigates the Triple Product Property (TPP), a central combinatorial condition in the Cohn‑Umans group‑theoretic framework for fast matrix multiplication. In this framework, a finite group G that contains three subsets S, T, U satisfying the TPP yields an embedding of matrix multiplication into the group algebra ℂ