The solutions of \(x^ d=1\) in finite groups
For \(G\) a finite group and \(d\) a positive integer, let \(\alpha(G, d)\) denote the number of elements \(x\in G\) with \(x^d= 1\). \textit{S. Chowla}, \textit{I. N. Herstein} and \textit{W. R. Scott} [Norske Vid. Selsk. Fordhdl. 25, 29-31 (1953; Zbl 0050.25605)] have shown the following identity for the symmetric groups \(S_n\): \[ \sum^\infty_{n= 0} {\alpha(S_n, d)\over n!} x^n= \exp\Biggl( \sum_{k|d} {x^k\over k}\Biggr). \] The author generalizes this result to obtain \[ \sum^\infty_{n= 0} {\alpha(G\wr S_n, d)\over n!} x^n= \exp\Biggl( \sum_{k|d} {|G|^{k- 1} \alpha(G, d/k)\over k} x^k\Biggr) \] for arbitrary finite groups \(G\). (Here, \(G\wr S_n\) is the wreath product of \(G\) with \(S_n\), i.e., the semidirect product \(G^n\ltimes S_n\), where \(S_n\) acts on \(G^n\) by permuting the factors.) He also gives similar formulas for \(G\wr A_n\) (\(A_n\) the alternating group) and for the Weyl groups \(W(D_n)\). Furthermore, the asymptotic behavior of the numbers \(\alpha(G\wr S_n, d)\) for \(n\to \infty\) is determined, generalizing a result of \textit{H. S. Wilf} [Bull. Am. Math. Soc., New Ser. 15, 228-232 (1986; Zbl 0613.05007)] on \(\alpha(S_n, d)\).
scientific article; zbMATH DE number 870158