Cycle lengths in finite groups and the size of the solvable radical

Abstract

We prove the following: For any $\rho \in \left(0,1\right)$, if a finite group $G$ has an automorphism with a cycle of length at least $\rho \cdot |G|$, then the index of the solvable radical $\mathrm{Rad}(G)$ in $G$ is bounded from above in terms of $\rho$, and such a condition is strong enough to imply solvability of $G$ if and only if $\rho >\frac{1}{10}$. Furthermore, considering, for exponents $e\in \left(0,1\right)$, the condition that a finite group $G$ have an automorphism with a cycle of length at least $|G{|}^e$, such a condition is strong enough to imply $|\mathrm{Rad}(G)|\to \infty$ for $|G|\to \infty$ if and only if $e>\frac{1}{3}$. We also prove similar results for a larger class of bijective self-transformations of finite groups, so-called periodic affine maps.

20 pages, 1 table, complete revision of the old version (from Jan 2015)

References in corpus (1)

• Classification of finite group automorphisms with a large cycle