On the Mِbius Number of the Subgroup Lattice of the Symmetric Group

In this paper, a formula is given for the Möbius numberμ(1, Sn) of the subgroup lattice of the symmetric groupSn. This formula involves the Möbius numbers of certain transitive subgroups ofSn. Whennhas at most two (not necessarily distinct) prime factors ornis a power of two, this formula is refined so that it involves only the Möbius numbers of certain primitive subgroups ofSn. Using the OʹNan-Scott Theorem, the classification of finite simple groups, and the refined formula, the exact value ofμ(1, Sn) is determined whennis prime, twice a prime, or a power of two. For certain primesp, |μ(1, S2p)|≠(2p)!/2. This result gives a negative answer to a question raised by Stanley.