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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 \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 $\mc P$.