LAUSR

Abstract Let σ = { σ i ∣ i ∈ I } {\sigma=\{\sigma_{i}\mid i\in I\}} be a partition of the set of all primes ℙ {\mathbb{P}} and G a finite group. A set ℋ {{\mathcal{H}}} of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 {\neq 1} of ℋ {{\mathcal{H}}} is a Hall σ i {\sigma_{i}} -subgroup of G for some i ∈ I {i\in I} and ℋ {\mathcal{H}} contains exactly one Hall σ i {\sigma_{i}} -subgroup of G for every i such that σ i ∩ π ⁢ ( G ) ≠ ∅ {\sigma_{i}\cap\pi(G)\neq\emptyset} . Let τ ℋ ( A ) = { σ i ∈ σ ( G ) ∖ σ ( A ) ∣ σ ( A ) ∩ σ ( H G ) ≠ ∅ \tau_{\mathcal{H}}(A)=\{\sigma_{i}\in\sigma(G)\setminus\sigma(A)\mid\sigma(A)% \cap\sigma(H^{G})\neq\emptyset for a Hall σ i {\sigma_{i}} -subgroup H of G } {G\}} . We say that a subgroup A of G is τ σ {\tau_{\sigma}} -permutable or τ σ {\tau_{\sigma}} -quasinormal in G with respect to ℋ {{\mathcal{H}}} if A ⁢ H x = H x ⁢ A {AH^{x}=H^{x}A} for all x ∈ G {x\in G} and all H ∈ ℋ {H\in\mathcal{H}} such that σ ⁢ ( H ) ⊆ τ ℋ ⁢ ( A ) {\sigma(H)\subseteq\tau_{\mathcal{H}}(A)} , and τ σ {\tau_{\sigma}} -permutable or τ σ {\tau_{\sigma}} -quasinormal in G if A is τ σ {\tau_{\sigma}} -permutable in G with respect to some complete Hall σ-set of G. We study G assuming that τ σ {\tau_{\sigma}} -quasinormality is a transitive relation in G.