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 .

Let m, n be positive integers, v a multilinear commutator word and w = v m . We prove that if G is a residually finite group in which all wvalues are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent. then x is a (left) n-Engel element. A group G is called n-Engel if all elements of G are n-Engel. It is a long-standing problem whether any n-Engel group is locally nilpotent. Following Zelmanov's solution of the restricted Burnside problem [16,17], Wilson proved that this is true if G is residually finite [14]. Later the second author showed that if in a residually finite group G all commutators [x 1 , . . . , x k ] are n-Engel, then the subgroup [x 1 , . . . , x k ] | x i ∈ G is locally nilpotent [10,11]. This suggests the following conjecture.Conjecture. Let w be a group-word and n a positive integer. Assume that G is a residually finite group in which all w-values are n-Engel. Then the corresponding verbal subgroup w(G) is locally nilpotent. Preliminary resultsGiven subgroups X and Y of a group G, we denote by X Y the smallest subgroup of G containing X and normalized by Y .Lemma 2.1. Let x and y be elements of a group G satisfying [x, n y m ] = 1, for some n, m ≥ 1. Then x y is finitely generated.Proof. Set X = x y m . Then X is finitely generated by [9, Exercise 12.3.6]. Since x y = X y i | i = 0, . . . , m − 1 , the lemma follows.Corollary 2.2. Let y be an element of a group G and H a finitely generated subgroup. If y m is Engel for some m ≥ 1, then H y is finitely generated.The following lemma is well-known. We supply the proof for the reader's convenience.

Let $G$, $H$ be groups. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. We prove that if $G$ and $H$ are groups that act compatibly on each other and such that the set of all tensors $T_{\otimes}(G,H)=\{g\otimes h \, : \, g \in G, \, h\in H\}$ is finite, then the non-abelian tensor product $G \otimes H$ is finite. In the opposite direction we examine certain finiteness conditions of $G$ in terms of similar conditions for the tensor square $G \otimes G$.Comment: The content of this paper improves and extends the main results of arXiv:1603.07003 [math.GR

Let G be a finite group with the property that if a, b are commutators of coprime orders, then |ab| = |a||b|. We show that G ′ is nilpotent.The following criterion of nilpotency of a finite group was established by B. Baumslag and J. Wiegold [1].Theorem 1. Let G be a finite group in which |ab| = |a||b| whenever the elements a, b have coprime orders. Then G is nilpotent.

We prove that the kth term of the lower central series of a finite group G is nilpotent if and only if |ab| = |a||b| for any γ k -commutators a, b ∈ G of coprime orders.2010 Mathematics Subject Classification. 20D30, 20D25.