On the Mobius number of the subgroup lattice of the symmetric group

摘要

In this paper, a formula is given for the Mobius number mu( l, S-n) of the subgroup lattice of the symmetric group S-n. This formula involves the Mobius numbers of certain transitive subgroups of S-n. When n has at most two (not necessarily distinct) prime factors or n is a power of two. this formula is refined so that it involves only the Mobius numbers of certain primitive subgroups of S-n. Using the O'Nan-Scott Theorem. the classification of finite simple groups, and the refined formula. the exact value of mu(l, S-n) is determined when n is prime, twice a prime, or a power of two. For certain primes p, mu(l, S-2p) not equal (2p)/2. This result gives a negative answer to a question raised by Stanley. (C) 1997 Academic Press.