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

In this paper, a formula is given for the Möbius numberμ(1, S_n) of the subgroup lattice of the symmetric groupS_n. This formula involves the Möbius numbers of certain transitive subgroups ofS_n. 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 ofS_n. Using the O'Nan-Scott Theorem, the classification of finite simple groups, and the refined formula, the exact value ofμ(1, S_n) is determined whennis prime, twice a prime, or a power of two. For certain primesp, μ(1, S_2p)≠(2p)!/2. This result gives a negative answer to a question raised by Stanley.