Let $M$ be an $S_n$-lattice (so it is free as an abelian group), and assume that $M$ is projective (i.e. direct summand of some $\mathbb Z[S_n]^m$). A theorem of Swan implies that $M$ is stably permutation, that is, $M\oplus \mathbb Z[E]=\mathbb Z[F]$, for some finite $S_n$-sets $E$ and $F$.

The theorems that go into the proof of this result are quite abstract and not constructive, I think. Is there a way to control $E$ and $F$? For example, can I always choose them so that no orbit has an $A_n$-stabilizer? Or such that every stabilizer is a $p$-group?


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