Abstract. We are looking for the smallest integer k > 1 providing the following characterization of the solvable radical R.G/ of any finite group G: R.G/ coincides with the collection of all g 2 G such that for any k elements a 1 ; a 2 ; : : : ; a k 2 G the subgroup generated by the elements g; a i ga 1 i , i D 1; : : : ; k, is solvable. We consider a similar problem of finding the smallest integer`> 1 with the property that R.G/ coincides with the collection of all g 2 G such that for any`elements b 1 ; b 2 ; : : : ; b`2 G the subgroup generated by the commutators OEg; b i , i D 1; : : : ;`, is solvable. Conjecturally, k D`D 3. We prove that both k and`are at most 7. In particular, this means that a finite group G is solvable if and only if every 8 conjugate elements of G generate a solvable subgroup.

Abstract. If G is a finite simple group of Lie type over a field containing more than 8 elements (for twisted groups l Xn(q l ) we require q > 8, except for 2 B 2 (q 2 ), 2 G 2 (q 2 ), and 2 F 4 (q 2 ), where we assume q 2 > 8), then G is the square of some conjugacy class and consequently every element in G is a commutator.

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Let G = G(K) where G is a simple and simply-connected algebraic group that is defined and quasi-split over a field K. We investigate properties of intersections of Bruhat cells BẇB of G with conjugacy classes C of G, in particular, we consider the question, when is BẇB ∩ C = ∅.