On regular G-grading

Abstract

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= {\Sigma}_{g in G} A_g that satisfies the following two conditions: (1) for every integer n>=1 and every n-tuple (g_1,g_2,...,g_n) in G^n, there are elements, a_i in A_{g_i}, i=1,...,n, such that a_1*a_2*...*a_n != 0. (2) for every g,h in G and for every a_g in A_g,b_h in A_h, we have a_{g}b_{h}=theta(g,h)b_{h}a_{g}. 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.