Suppose that a finite group G admits a Frobenius group of automorphisms with kernel F and complement H such that the fixed-point subgroup of F is trivial: . In this situation various properties of G are shown to be close to the corresponding properties of . By using Clifford's theorem it is proved that the order is bounded in terms of and , the rank of G is bounded in terms of and the rank of , and that G is nilpotent if is nilpotent. Lie ring methods are used for bounding the exponent and the nilpotency class of G in the case of metacyclic . The exponent of G is bounded in terms of and the exponent of by using Lazard's Lie algebra associated with the Jennings–Zassenhaus filtration and its connection with powerful subgroups. The nilpotency class of G is bounded in terms of and the nilpotency class of by considering Lie rings with a finite cyclic grading satisfying a certain `selective nilpotency' condition. The latter technique also yields similar results bounding the nilpotency class of Lie rings and algebras with a metacyclic Frobenius group of automorphisms, with corollaries for connected Lie groups and torsion-free locally nilpotent groups with such groups of automorphisms. Examples show that such nilpotency results are no longer true for non-metacyclic Frobenius groups of automorphisms

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Abstract. We say that a group G is almost Engel if for every g ∈ G there is a finite set E (g) such that for every x ∈ G all sufficiently long commutators [...[[x, g](Thus, Engel groups are precisely the almost Engel groups for which we can choose E (g) = {1} for all g ∈ G.)We prove that if a compact (Hausdorff) group G is almost Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound |E (g)| m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson-Zelmanov theorem saying that Engel profinite groups are locally nilpotent.

An Engel sink of an element [Formula: see text] of a group [Formula: see text] is a set [Formula: see text] such that for every [Formula: see text] all sufficiently long commutators [Formula: see text] belong to [Formula: see text]. (Thus, [Formula: see text] is an Engel element precisely when we can choose [Formula: see text].) It is proved that if every element of a compact (Hausdorff) group [Formula: see text] has a countable (or finite) Engel sink, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. This settles a question suggested by J. S. Wilson.

Abstract. Every finite group G has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length λ(G) as the minimum number of nonsoluble factors in a series of this kind. Upper bounds for λ(G) appear in the study of various problems on finite, residually finite, and profinite groups. We prove that λ(G) is bounded in terms of the maximum 2-length of soluble subgroups of G, and that λ(G) is bounded by the maximum Fitting height of soluble subgroups. For an odd prime p, the non-p-soluble length λ p (G) is introduced, and it is proved that λ p (G) does not exceed the maximum p-length of p-soluble subgroups. We conjecture that for a given prime p and a given proper group variety V the non-psoluble length λ p (G) of finite groups G whose Sylow p-subgroups belong to V is bounded. In this paper we prove this conjecture for any variety that is a product of several soluble varieties and varieties of finite exponent. As an application of the results obtained, an error is corrected in the proof of the main result of the second author's paper "Multilinear commutators in residually finite groups", Israel J. Math. 189 (2012), 207-224.

In 1954 B. H. Neumann discovered that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G ′ is finite. Later (in 1957) Wiegold found an explicit bound for the order of G ′ . We study groups in which the conjugacy classes containing commutators are finite with bounded size. We obtain the following results.Let G be a group and n a positive integer. If |x G | ≤ n for any commutator x ∈ G, then the second derived group G ′′ is finite with n-bounded order.If |x G ′ | ≤ n for any commutator x ∈ G, then the order of γ 3 (G ′ ) is finite and n-bounded.

Abstract. A group-word w

Let g be an element of a finite group G. For a positive integer n, let E n (g) be the subgroup generated by all commutators [...[[x, g], g], . . . , g] over x ∈ G, where g is repeated n times. By Baer's theorem, if E n (g) = 1, then g belongs to the Fitting subgroup F (G). We generalize this theorem in terms of certain length parameters of E n (g). For soluble G we prove that if, for some n, the Fitting height of E n (g) is equal to k, then g belongs to the (k + 1)th Fitting subgroup F k+1 (G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h * (H) of a finite group H is the least number h such that F * h (H) = H, where F * 0 (H) = 1, and F * i+1 (H) is the inverse image of the generalized Fitting subgroup F * (H/F * i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of E n, where f (k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(E n (g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.