On the Generalized Burnside Ring with Respect to the Young Subgroups of the Symmetric Group

We determine the number of blocks of the generalized Burnside ring of the symmetric group Sn with respect to the Young subgroups of Sn over a field of characteristic p. Let kSn be a group algebra of Sn over a field k of characteristic p > 0 and (kSn)(p) the Grothendieck ring of kSn over p-local integers. Then, as a corollary of the theorem, we have that F (kSn)(p) is semisimple, where F is any field of characteristic p. It is well known that the result holds for an arbitrary finite group, but our approach to the result is remarkable.