Let *σ* = {*σ*_{i} ∣ *i* ∈ *I*} be a partition of the set of all primes P and *G* a finite group. A set *H* of subgroups of *G* is said to be a *complete Hall σ-set* of *G* if every member ≠ 1 of *H* is a Hall *σ*_{i}-subgroup of *G* for some *i* ∈* I* and *H* contains exactly one Hall *σ*_{i}-subgroup of *G* for every *i* such that *σ*_{i} ∩ *π*(*G*) ≠ ∅.
Let *τ*_{H}(*A*) = {*σ*_{i} ∈ *σ*(*G*) \ *σ*(*A*) ∣ *σ*(*A*) ∩ *σ*(*H*^{G}) ≠ ∅ for a Hall *σ*_{i}-subgroup *H* of *G*}. We say that a subgroup *A* of *G* is * τ*_{σ}-permutable or * τ*_{σ}-quasinormal in *G* with respect to H if *AH*^{x} = *H*^{x} A for all *x* ∈ *G* and all *H* ∈ *H* such that *σ*(*H*) ⊆ *τ*_{H}(*A*), and * τ*_{σ}-permutable or * τ*_{σ}-quasinormal in *G* if *A* is * τ*_{σ}-permutable in *G* with respect to some complete Hall *σ*-set of *G*.

We study *G* assuming that *τ*_{σ}-quasinormality is a transitive relation in *G*.

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