Character polynomials and the Möbius function
This elegant paper concerns the Möbius function \(\mu\) on the partially ordered set of all subgroups of a finite group \(G\). Assume \(G\) is acting by permutations on some set \(\Omega\). Then the permutation structure of the action of \(G\) on the permutation modules \(\mathbb{F}_q \Omega\) over \(\mathbb{F}_q\) can be described by some polynomials in \(q\) [see \textit{H. Pahlings} and \textit{W. Plesken}, J. Reine Angew. Math. 380, 178-195 (1987; Zbl 0613.20008), or \textit{A. Turull}, Proc. Am. Math. Soc. 103, No. 3, 463-467 (1988; Zbl 0651.20012)]. The present paper uses these polynomials to obtain relations among the values of the \(\mu\) function on the transitive subgroups of \(G\). Particular attention is paid to the symmetric and alternating groups. It is proved, using the classification of finite simple groups, that if \(p\) is any prime, then \(\mu(1,S_p)=(-1)^{p+1}p!/2\). For many such \(p\), it is also shown that \(\mu(1,A_p)=-\mu(1,S_p)\).
scientific article; zbMATH DE number 800592
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