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 $$s, s' \in S$$, $$t, t' \in T$$, $$u, u' \in U$$ 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.