Fonction de Möbius d'un groupe fini et anneau de Burnside. (Möbius function of a finite group and the Burnside ring) (Q1075424)

This paper gives various relationships between the structure and idempotents of the Burnside ring \(\Omega\) (G) of a finite group G and its lattice S(G) of subgroups, in particular the Möbius function of S(G). Crapo's formula for the Möbius function of a lattice in terms of the complements of a fixed point of the lattice is reproved for S(G). The Möbius function for nilpotent and soluble groups is set down explicitly, the latter in terms of the number of complements of the terms of a principal series for the group. If \(H\leq G\), and if \(| G:G'|_ o\) is the product of the distinct prime factors of \(| G:G'|\), then \(| G:G'|_ o\cdot \mu (H,G)\) is a multiple of \(| N_ G(H):H|\); and this last result is best possible.