Triple product property

In abstract algebra, the triple product property is an identity satisfied in some groups. Let $G$ be a non-trivial group. Three nonempty subsets $S,T,U\subset G$ are said to have the triple product property in $G$ if for all elements $s,s^{\prime }\in S$, $t,t^{\prime }\in T$, $u,u^{\prime }\in U$ it is the case that

$s^{\prime }{s}^{-1}{t}^{\prime }{t}^{-1}{u}^{\prime }{u}^{-1}=1\Rightarrow s^{\prime }=s,t^{\prime }=t,u^{\prime }=u$

where $1$ is the identity of $G$.

It plays a role in research of fast matrix multiplication algorithms.

References

• Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. arXiv:math.GR/0307321. Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 11–14 October 2003, Cambridge, MA, IEEE Computer Society, pp. 438–449.