Let {θn}∞ n=1 be a sequence of words. If there exists a positive integer n such that θm(G) = 1 for every m ≥n,then we say that G satisfies (*) and denote the class of all groups satisfying (*) by X{θn}∞ n=1. If for every proper subgroup K of G, K ∈ X{θn}∞ n=1 but G /∈ X{θn}∞ n=1, then we call G a minimal non-X{θn}∞ n=1-group. Assume that G is an infinite locally finite group with trivial center and θi(G) = G for all i ≥ 1. In this case we mainly prove that there exists a positive integer t such that for every proper normal subgroup N of G, either θt(N) = 1 or θt(CG(N)) = 1. We also give certain useful applications of the main result.