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 ( x 1 − 1 g 1 x 1 ) ⋯ ( x m − 1 g m x m ) = 1 , where g i is in M and x i is equal to a , b , or c for all i between 1 and m , then x i = x i + 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. [ABSTRACT FROM AUTHOR]