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 \lineq over \Q, of cardinality \leq 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 \mc P of primes,
Sn-sets linearly equivalent over a field k if and only if the
characteristic of k lies in \mc P.Comment: v2: fixed proof of Lemma 2.