Triple product property

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

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

s ′ s − 1 t ′ t − 1 u ′ u − 1 = 1 ⇒ s ′ = s , t ′ = t , u ′ = u {\displaystyle s's^{-1}t't^{-1}u'u^{-1}=1\Rightarrow s'=s,t'=t,u'=u}

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


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