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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
where 1 {\displaystyle 1} is the identity of G {\displaystyle G}.
It plays a role in research of fast matrix multiplication algorithms.