A solvable version of the Baer–Suzuki theorem

Abstract

Suppose that G is a finite group and x ∈ G has prime order p > 5. Then x is contained in the solvable radical of G, O ∞ (G), if (and only if) 〈x, x g 〉 is solvable for all g ∈ G. If G is an almost simple group and x ∈ G has prime order p > 5, then this implies that there exists g ∈ G such that 〈x, x g 〉 is not solvable. In fact, this is also true when p = 3 with very few exceptions, which are described explicitly.