#### Abstract

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*-subgroup of

_{i}*G*for some

*i*∈

*I*and

*H*contains exactly one Hall

*σ*-subgroup of

_{i}*G*for every

*i*such that

*σ*∩

_{i}*π*(

*G*) ≠ ∅.

Let *τ _{H}*(

*A*) = {

*σ*∈

_{i}*σ*(

*G*) \

*σ*(

*A*) ∣

*σ*(

*A*) ∩

*σ*(

*H*) ≠ ∅ for a Hall

^{G}*σ*-subgroup

_{i}*H*of

*G*}. We say that a subgroup

*A*of

*G*is

*τ*or

_{σ}-permutable*τ*if

_{σ}-quasinormal in*G*with respect to H*AH*=

^{x}*H*for all

^{x}A*x*∈

*G*and all

*H*∈

*H*such that

*σ*(

*H*) ⊆

*τ*(

_{H}*A*), and

*τ*or

_{σ}-permutable*τ*in

_{σ}-quasinormal*G*if

*A*is

*τ*in

_{σ}-permutable*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*.

#### Document Type

Article

#### Publication Date

9-1-2017

#### Digital Object Identifier (DOI)

https://doi.org/10.1515/jgth-2017-0016

#### Repository Citation

Beidleman, James C. and Skiba, Alexander N., "On *τ _{σ}*-Quasinormal Subgroups of Finite Groups" (2017).

*Mathematics Faculty Publications*. 29.

https://uknowledge.uky.edu/math_facpub/29

## Notes/Citation Information

Published in

Journal of Group Theory, v. 20, issue 5, p. 955-969.© de Gruyter 2017

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