Small linearly equivalent G-sets and a construction of Beaulieu

Two G-sets (G a finite group) are called linearly equivalent over a commutative ring k if the permutation representations k[X] and k[Y] are isomorphic as modules over the group algebra kG. Pairs of linearly equivalent non-isomorphic G-sets have applications in number theory and geometry. We characterize the groups G for which such pairs exist for any field, and give a simple construction of these pairs. If k is Q, these are precisely the non-cyclic groups. For any non-cyclic group, we prove that there exist G-sets which are non-isomorphic and linearly equivalent over Q, of cardinality ⩽3(#G)/2. Also, we investigate a construction of P. Beaulieu which allows us to construct pairs of transitive linearly equivalent Sn-sets from arbitrary G-sets for an arbitrary group G. We show that this construction works over all fields and use it construct, for each finite set P of primes, Sn-sets linearly equivalent over a field k if and only if the characteristic of k lies in P.