Non-noetherian groups and primitivity of their group algebras

We prove that the group algebra KG of a group G over a field K is primitive, provided that G has a non-abelian free subgroup with the same cardinality as G, and that G satisfies the following condition (⁎): for each subset M of G consisting of a finite number of elements not equal to 1, and for any positive integer m, there exist distinct a, b, and c in G so that if (x1−1g1x1)⋯(xm−1gmxm)=1, where gi is in M and xi is equal to a, b, or c for all i between 1 and m, then xi=xi+1 for some i. This generalizes results of [1,9,17], and [18], and proves that, for every countably infinite group G satisfying (⁎), KG is primitive for any field K. We use this result to determine the primitivity of group algebras of one relator groups with torsion.