We deduce some elementary pairwise disjointness and semi-disjointness
conditions on triples of subsets in arbitrary groups satisfying the so-called
triple product property (TPP) as originally defined by H. Cohn and C. Umans in
2003. This property TPP for a triple of group subsets, called a TPP triple,
allows the group to "realize" matrix multiplication of dimensions the sizes of
the subsets, with the subsets acting as indexing sets for input matrices which
are embedded into the regular algebra of the group.