Almost equal group multiplications

In a recent paper entitled "A commutative analogue of the group ring" we introduced, for each finite group (G, ), a commutative graded Z-algebra R-(G,R- ) which has a close connection with the cohomology of (G, ). The algebra R-(G,R-) is the quotient of a polynomial algebra by a certain ideal I(G) and it remains a fundamental open problem whether or not the group multiplication on G can always be recovered uniquely from the ideal I-(G,I- ()).

Suppose now that (G. x) is another group with the same underlying set G and identity element e is an element of G such that I-(G,I- ()) = I-(G,I-x) Then we show here that the multiplications and x are at least "almost equal'' in a precise sense which renders them indistinguishable in terms of most of the standard group theory constructions. In particular in many cases (for example if (G.) is Abelian or simple) this implies that the two multiplications are actually equal as was claimed in the previously cited paper