Triple product property

In abstract algebra, the triple product property is an identity satisfied in some groups. Let [math]\displaystyle{ G }[/math] be a non-trivial group. Three nonempty subsets [math]\displaystyle{ S, T, U \subset G }[/math] are said to have the triple product property in [math]\displaystyle{ G }[/math] if for all elements [math]\displaystyle{ s, s' \in S }[/math], [math]\displaystyle{ t, t' \in T }[/math], [math]\displaystyle{ u, u' \in U }[/math] it is the case that

[math]\displaystyle{ s's^{-1}t't^{-1}u'u^{-1} = 1 \Rightarrow s' = s, t' = t, u' = u }[/math]

where [math]\displaystyle{ 1 }[/math] is the identity of [math]\displaystyle{ G }[/math].

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.