Search and test algorithms for Triple Product Property triples

TL;DR

This paper introduces new characterizations and algorithms for testing the Triple Product Property in groups, aiding the search for group-theoretic methods to improve matrix multiplication efficiency.

Contribution

It provides novel characterizations of TPP, implements TPP test algorithms in GAP, and explores subgroup TPP triples, advancing the computational tools for matrix multiplication research.

Findings

All known TPP tests are described and compared.

Search restrictions to nonnormal subgroups are established.

Brute-force search results for small groups and specific classes are presented.

Abstract

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 $PSL(2,q)$.