  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 $S_n$-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,
$S_n$-sets linearly equivalent over a field $k$ if and only if the
characteristic of $k$ lies in $\mc P$.
