ON SCHUR 3-GROUPS

Let G be a finite group. An S-ring A over G is a subring of the group ring ZG that has a linear basis associated with a special partition of G. About 40 years ago R. Poschel suggested the problem which can be formulated as follows: for which group G every S-ring A over it is schurian, i.e. the partition of G corresponding to A consists of the orbits of the one point stabilizer of a permutation group in Sym(G) that contains a regular subgroup isomorphic to G. The main result of the paper says that such G can not be non-abelian p-group, where p is an odd prime. In fact, modulo known results, it was sufficient to show that for every n >= 3 there exists a non-schurian S-ring over the group M-3n = < a, b vertical bar a(3n-1) = b(3) = e, a(b) = a(3n-2+1)>.