Finite groups admitting some coprime operator groups.

Let G be a finite group, with a finite operator group A, satisfying the following conditions: (1) (|G|, |A|) = 1; (2) there exists a natural number m such that for any α, β ∈ A we have: [ C_G (α), C_G (β), . .<-m-> . , C_G (β) ] = {1}; (3) A is not cyclic. We prove the following: (1) If the exponent n of A is square-free, then G is nilpotent and its class is bounded by a function depending only on m and λ(n) (= n). (2) If Z(A) = {1} and A has exponent n, then G is nilpotent and its class is bounded by a function depending only on m and λ(n).