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

✍ Scribed by John Shareshian

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 460 KB
• Volume: 78
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2762

✦ Synopsis

In this paper, a formula is given for the Mo bius number +(1, S n ) of the subgroup lattice of the symmetric group S n . This formula involves the Mo bius 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 Mo bius 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 +(1, S n ) is determined when n is prime, twice a prime, or a power of two. For certain primes p, |+(1, S 2p )|{(2p)!Â2. This result gives a negative answer to a question raised by Stanley.