On regular G-gradings

Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A =(formula present)that satisfies the following two conditions: (1) for every integer n ≥ 1 and every n-tuple (g1, g2, . . . , gn) ∈ Gⁿ, there are elements, a_i∈ A_{gi ,}i = 1, . . . ,n, such that Пⁿ₁a_i≠ = 0; (2) for every g, h ∈ G and for every a_g∈ A_g, b_h∈ A_h, we have a_gb_h= θ_g,hb_ha_gfor some nonzero scalar θ_g,h. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.