Triple product property

In abstract algebra, the triple product property is an identity satisfied in some groups.

Let ${\displaystyle G}$ be a non-trivial group. Three nonempty subsets ${\displaystyle S,T,U\subset G}$ are said to have the triple product property in ${\displaystyle G}$ if for all elements ${\displaystyle s,s'\in S}$, ${\displaystyle t,t'\in T}$, ${\displaystyle u,u'\in U}$ it is the case that

${\displaystyle s's^{-1}t't^{-1}u'u^{-1}=1\Rightarrow s'=s,t'=t,u'=u}$

where ${\displaystyle 1}$ is the identity of ${\displaystyle G}$.

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.