A solvable version of the Baer--Suzuki Theorem
Abstract
Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,xg> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then this implies that there exists g in G such that <x,xg> is not solvable. In fact, this is also true when p=3 with very few exceptions, which are described explicitly.
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