Characterizations of the solvable radical

Abstract

We prove that there exists a constant k k with the property: if C \mathcal {C} is a conjugacy class of a finite group G G such that every k k elements of C \mathcal {C} generate a solvable subgroup, then C \mathcal {C} generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take k = 4 k=4 . We also present proofs that do not use the Classification Theorem. The most direct proof gives a value of k = 10 k=10 . By lengthening one of our arguments slightly, we obtain a value of k = 7 k=7 .