Characterizations of the Solvable Radical

Flavell, Paul; Guest, Simon; Guralnick, Robert

Characterizations of the Solvable Radical

Authors

Flavell, Paul

Guest, Simon

Guralnick, Robert

Description

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