ON FINITE FACTORIZED GROUPS WITH TX-SUBNORMAL SUBGROUPS

Recall that a subgroup H is called T-subnormal in G if either H = G or there is a chain of subgroups H = H-0 = H-1 = ... =H-n = G such that |H-i : Hi-1| ? T for all i. Let X be a normal subgroup of the group G and let T be the set of natural numbers satisfying the condition (A). We introduce the following definition: A subgroup H of the group G is called a TX-subnormal subgroup if H is T-subnormal in HX. Moreover, we study factorizable groups G = AB with TX-subnormal factors A and B. Under additional restrictions imposed on A, B, T, and X, we obtain new sufficient conditions for the partial solubility and supersolubility of the analyzed group G.