  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 $\operatorname{Rad}(G)$ 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 $|\operatorname{Rad}(G)|\to\infty$ 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.
