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

  • Peter M. Neumann (a1)

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,dG.

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 s1s2s3v2G∣.

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Copyright

References

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[1]Cohn, Henry and Umans, Christopher, ‘A group-theoretic approach to fast matrix multiplication’, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS’03) (IEEE, 2003) 438449.
[2]Cohn, Henry, Kleinberg, Robert, Szegedy, Balázs and Umans, Christopher, ‘Group-theoretic algorithms for matrix multiplication’, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) (IEEE, 2005) 379388.
[3]Curtis, Charles W., Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer, AMS–LMS Series on History of Mathematics 15 (American Mathematical Society, 1999).
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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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