Integral representations of groups of square-free order

Abstract

We describe the structure of the integral group ring Z G, when G has square-free order, as a subdirect sum of hereditary orders in skew group algebras. From this we deduce the structure of all genera of Z G-lattices. Our principal applications are the following (for groups G of square-free order). (i) We determine those G whose Z G-lattices satisfy uniqueness of the number of indecomposable summands. (ii) We determine those G whose Z G-lattices are direct sums of left ideals, (iii) For those G whose Z G-lattices are not direct sums of left ideals, we show that indecomposable Z G-lattices can be much larger than the ring Z G itself, despite the fact that Z G is of finite representation type and, over the p-adic completions of Z G, lattices always become direct sums of left ideals. (iv) We show that the ring structure of Q G determines the group G up to isomorphism.