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

作者：Neumann Peter M^*

来源：LMS Journal of Computation and Mathematics, 2011, 14: 232-237.

DOI：10.1112/S1461157010000288

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 (S-1, S-2 S-3) is a TPP triple in a finite group G, then so is (dS(1)a, dS(2)b, dS(3)c) for any a, b, c, d is an element of G.
Let s(i) := vertical bar S-i vertical bar for 1 <= i <= 3. First we prove the inequality s(1)(s(2) + s(3) - 1) <= vertical bar G vertical bar 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 s(1)s(2)s(3) <= v(2)vertical bar G vertical bar.

• 出版日期2011