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,⋅) which has a close connection with the cohomology of (G,⋅). The algebra R(G,⋅) 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,⋅).Suppose now that (G,×) is another group with the same underlying set G and identity element e∈G such that I(G,⋅)=I(G,×). Then we show here that the multiplications ⋅ and × 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.