Search and test algorithms for triple product property triples

Abstract

In 2003 Cohn and Umans introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group satisfying the triple product property (TPP) as a means to bound the exponent of matrix multiplication. We present two new characterizations of the TPP, which are used for theoretical considerations and for TPP test algorithms. We describe the algorithms for all known TPP tests and present the runtime differences between their GAP implementations. We prove that the search for non-trivial-sized TPP triples of subgroups of a given group can be restricted to the set of its non-normal subgroups, and apply this, together with other preconditions, to describe brute-force search algorithms for largest-sized TPP triples of subgroups and subsets. In addition we present the results of the subset brute-force search for all groups of order up to 32 and selected results of the subgroup brute-force search for 2-groups, and . Our results for the groups and suggest tentative answers to certain questions posed by Cohn and Umans.