Hall's relations in finite groups.
Let \(G\) be a finite group, let \(n\) be a fixed positive integer and let \(\alpha\in\Aut(G)\) be such that the order of \(\alpha\) divides \(n\). Let \(l_n(G,\alpha)\) denote the number of elements of \(G\) satisfying the equality \(xx^\alpha x^{\alpha^2}\cdots x^{\alpha^{n-1}}=1\). \textit{Ph. Hall} [Proc. Lond. Math. Soc., II. Ser. 40, 468-480 (1935), 481-501 (1936; Zbl 0015.20201)] extended a well-known result of Frobenius by showing that \((n,|G|)\) divides \(l_n(G,\alpha)\). In the article under review, the authors show that certain arithmetic properties of the number \(l_n(G,\alpha)\) have a strong impact on the internal structure of \(G\). Specifically, the main result asserts that if the prime \(p\) divides \((n,|G|)\) and if \((pn,|G|)\) does not divide \(l_n(G,\alpha)\), then a Sylow \(p\)-subgroup of \(G\) is either cyclic (when \(p\) is odd), or is exceptional (when \(p=2\)). Here ``exceptional'' means cyclic, dihedral, generalized quaternion, or quasi-dihedral.
scientific article; zbMATH DE number 2037094