# Triple product property

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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 $s,s'\in S$ , $t,t'\in T$ , $u,u'\in U$ it is the case that

$s's^{-1}t't^{-1}u'u^{-1}=1\Rightarrow s'=s,t'=t,u'=u$ where $1$ is the identity of $G$ .

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

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• 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. 438449.