Bounding the size of permutation groups and complex linear groups of odd order. (Q655394)

Let \(G\) be a finite group of odd order for which \(p\) is the least prime dividing \(|G|\), and let \(F(G)\) denote the Fitting subgroup. The main results of this paper are the following: (a) if \(G\) is a permutation group of degree \(n\), then \(|G|\leq\alpha(p)^{n-1}\) where \(\alpha(3):=3^{1/2}\), \(\alpha(5):=5^{1/4}\) and \(\alpha(p):=[p(2p+1)]^{1/(2p)}\) for \(p>5\); and (b) if \(G\leq\text{GL}(n,\mathbb C)\) then \(|G:F(G)|\leq\alpha(p)^{n-1}\) and \(G\) has a normal Abelian subgroup \(A\) for which \(|G:A|\leq\beta(p)^{n-1}\) where \(\beta(p):=[p(2p+1)^2]^{1/(2p)}\) if \(p\equiv 2\pmod 3\) and \(\beta(p):=[p(2p-1)^2]^{1/(2p-2)}\) otherwise. A corollary gives inequalities between \(|G/F(G):F(G/F(G))|\), \(|G:\Phi(G)|\) and \(|F(G):\Phi(G)|\). \{It is known (see, for example, Section 5.8 of \textit{J. D. Dixon} and \textit{B. Mortimer}, Permutation groups. [Graduate Texts in Mathematics 163. New York: Springer-Verlag (1996; Zbl 0951.20001)]) that exponential bounds similar to (a) hold for many classes of permutation groups.\}