Communicated by M. Sapir MSC: Primary: 20D45 Secondary: 17B70; 20D15; 20F40 a b s t r a c t Suppose that a finite group G admits a Frobenius group of automorphisms FH of coprime order with kernel F and complement H such thatis repeated k times, then G is nilpotent of class bounded in terms of c, k and |H| only. It is also proved that if F is abelian of rank at least three and C G (a) is nilpotent of class at most d for every a ∈ F \ {1}, then G is nilpotent of class bounded in terms of c, d and |H|. The proofs are based on results on graded Lie rings.
The following theorem is proved. Let m, k and n be positive integers. There exists a number η = η(m, k, n) depending only on m, k and n such that if G is any residually finite group satisfying the condition that the product of any η commutators of the form [x m , y 1 , . . . , y k ] is of order dividing n, then the verbal subgroup of G corresponding to the word w = [x m , y 1 , . . . , y k ] is locally finite.2010 Mathematics subject classification: primary 20E10; secondary 20E26, 20F40, 20F50.
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