Stability of the centers of group algebras of general affine groups GA_n(q)

The general affine group GA_n(q) consisting of invertible affine transformations of an affine space of codimension one in the vector space F_qⁿ over a finite field F_q, can be viewed as a subgroup of the general linear group GL_n(q) over F_q. In the article, we define the type of an element of GA_n(q), and show that two elements are conjugate if and only if they have the same type. Then the center A_n(q) of the integral group algebra Z[GA_n(q)] is proved to be a filtered algebra via the length function defined in terms of the reflections belonging to GA_n(q). We show in the associated graded algebras G_n(q) the structure constants with respect to the basis consisting of the conjugacy class sums are independent of n. The structure constants in G_n(q) are further shown to contain the structure constants in the graded algebras introduced by the first author and Wang for GL_n(q) as special cases. The stability leads to a universal stable center G(q) with positive integer structure constants only depending on q which governs the algebras G_n(q) for all n.