Abstract. Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) 5, then G is isomorphic to one of the groups A 5 , A 6 , P SL(2, 7) or P SL(2, 8). We also consider the case when ω(G) = 6 and show that if G is a nonsolvable finite group with ω(G) = 6, then either G ≃ P SL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≃ A 5 .

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We prove that in a self-similar wreath product of abelian groups G = BwrX, if X is torsion-free then B is torsion of finite exponent. Therefore, in particular, the group ZwrZ cannot be self-similar Furthemore, we prove that if L is a self-similar abelian group then L ω wrC 2 is also self-similar. We thank the referee for suggestions which improved the original text.

LetGbe a finite group with the property that if{a,b}are powers of{\delta_{1}^{*}}-commutators such that{(|a|,|b|)=1}, then{|ab|=|a||b|}. We show that{\gamma_{\infty}(G)}is nilpotent.

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