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Fonction de Möbius d’un groupe fini et anneau de Burnside. (Möbius function of a finite group and the Burnside ring). (French) Zbl 0592.20004

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.
Reviewer: S.B.Conlon

MSC:

20C15 Ordinary representations and characters
20C10 Integral representations of finite groups
20D30 Series and lattices of subgroups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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