James C. Beidleman, Alexander N. Skiba
May 17, 2017

### Abstract

Let σ={σi∣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 σi{\sigma_{i}}-subgroup of G for some i∈I{i\in I} and ℋ{\mathcal{H}} contains exactly one Hall σi{\sigma_{i}}-subgroup of G for every i such that σi∩π(G)≠∅{\sigma_{i}\cap\pi(G)\neq\emptyset}. Let τℋ(A)={σi∈σ(G)∖σ(A)∣σ(A)∩σ(HG)≠∅\tau_{\mathcal{H}}(A)=\{\sigma_{i}\in\sigma(G)\setminus\sigma(A)\mid\sigma(A)% \cap\sigma(H^{G})\neq\emptyset for a Hall σi{\sigma_{i}}-subgroup H of G}{G\}}. We say that a subgroup A of G is τσ{\tau_{\sigma}}-permutable or τσ{\tau_{\sigma}}-quasinormal in G with respect to ℋ{{\mathcal{H}}} if AHx=HxA{AH^{x}=H^{x}A} for all x∈G{x\in G} and all H∈ℋ{H\in\mathcal{H}} such that σ(H)⊆τℋ(A){\sigma(H)\subseteq\tau_{\mathcal{H}}(A)}, and τσ{\tau_{\sigma}}-permutable or τσ{\tau_{\sigma}}-quasinormal in G if A is τσ{\tau_{\sigma}}-permutable in G with respect to some complete Hall σ-set of G . We study G assuming that τσ{\tau_{\sigma}}-quasinormality is a transitive relation in G .