High order elements and number of roots of unity in a finite group (Q1210152)

Let \(G\) be a finite group and let \(q_ 0,q_ 1,\dots,q_ h\) \((q_ 0 < q_ 1 < \dots < q_ h)\) be the prime divisors of \(| G|\). The author proves, by a combination of group-theoretic and number-theoretic methods, that \[ \iota_ G < {1 - 1/q_ 0 +\varepsilon(\gamma_ G)\over h!\log q_ 0\dots\log q_ h} \gamma_ G \log^ h\gamma_ G. \] Here \(\gamma_ G = \max_{d\mid | G|}\{\gamma\mid x^ d = 1\) has \(\gamma d\) solutions in \(G\}\) is the multiplicity of \(G\), and \(\iota_ G = \min_{x\in G}[G : \langle x\rangle]\) is the cocyclicity of \(G\), and \(\varepsilon(\gamma_ G)\to 0\) as \(\gamma_ 0 \to \infty\).