Abstract
A subgroup A of a group G is said to be X-permutable with another subgroup B in G, where ∅ ≠ X ⊆ G, if there exists some element x ∈ X such that ABx=BxA. In this paper, the solubility and supersolubility of finite groups are described by X-permutability of the Hall subgroups and their subgroups, in addition, the well known theorem of Schur-Zassenhaus in finite group is generalized.
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