LMS Journal of Computation and Mathematics

Table of Contents - Volume 14 - 2011  

Research Article

A note on the triple product property for subsets of finite groups

Peter M. Neumanna1

a1 The Queen’s College, Oxford OX1 4AW, United Kingdom (email: peter.neumann@queens.ox.ac.uk)

Abstract

The triple product property (TPP) for subsets of a finite group was introduced by Henry Cohn and Christopher Umans in 2003 as a tool for the study of the complexity of matrix multiplication. This note records some consequences of the simple observation that if (S1,S2,S3) is a TPP triple in a finite group G, then so is (dS1a,dS2b,dS3c) for any a,b,c,dxs2208G.

Let si:=∣Si∣ for 1≤i≤3. First we prove the inequality s1(s2+s3−1)≤∣G∣ and show some of its uses. Then we show (something a little more general than) that if G has an abelian subgroup of index v, then s1s2s3 ≤v2 ∣G∣.

(Received June 17 2010)

(Revised December 13 2010)

(Online publication September 2011)

2000 Mathematics Subject Classification

  • 20D60 (primary);
  • 68R05 (secondary)