# On $$\Pi$$ Π -quasinormal subgroups of finite groups

On $$\Pi$$ Π -quasinormal subgroups of finite groups Let $$\sigma =\{\sigma _{i} | i\in I\}$$ σ = { σ i | i ∈ I } be some partition of the set of all primes $$\mathbb {P}$$ P and $$\Pi$$ Π a non-empty subset of the set $$\sigma$$ σ . A set $$\mathcal{H}$$ H of subgroups of a finite group G is said to be a complete Hall $$\Pi$$ Π -set of G if every member $$\ne 1$$ ≠ 1 of $$\mathcal{H}$$ H is a Hall $$\sigma _{i}$$ σ i -subgroup of G for some $$\sigma _{i}\in \Pi$$ σ i ∈ Π and $$\mathcal{H}$$ H contains exactly one Hall $$\sigma _{i}$$ σ i -subgroup of G for every $$\sigma _{i}\in \Pi$$ σ i ∈ Π such that $$\sigma _i\cap \pi (G)\ne \emptyset$$ σ i ∩ π ( G ) ≠ ∅ . A subgroup H of G is called $$\Pi$$ Π -permutable or $$\Pi$$ Π -quasinormal in G if G possesses a complete Hall $$\Pi$$ Π -set $$\mathcal{H}$$ H such that $$AH^{x}=H^{x}A$$ A H x = H x A for all $$H\in \mathcal{H}$$ H ∈ H and $$x\in G$$ x ∈ G . We study the embedding properties of H under the hypothesis that H is $$\Pi$$ Π -permutable in G. Some well-known results are generalized. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

# On $$\Pi$$ Π -quasinormal subgroups of finite groups

, Volume 185 (3) – Dec 17, 2016
11 pages

/lp/springer_journal/on-pi-quasinormal-subgroups-of-finite-groups-h6GVg0ghM3
Publisher
Springer Vienna
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-016-1007-9
Publisher site
See Article on Publisher Site

### Abstract

Let $$\sigma =\{\sigma _{i} | i\in I\}$$ σ = { σ i | i ∈ I } be some partition of the set of all primes $$\mathbb {P}$$ P and $$\Pi$$ Π a non-empty subset of the set $$\sigma$$ σ . A set $$\mathcal{H}$$ H of subgroups of a finite group G is said to be a complete Hall $$\Pi$$ Π -set of G if every member $$\ne 1$$ ≠ 1 of $$\mathcal{H}$$ H is a Hall $$\sigma _{i}$$ σ i -subgroup of G for some $$\sigma _{i}\in \Pi$$ σ i ∈ Π and $$\mathcal{H}$$ H contains exactly one Hall $$\sigma _{i}$$ σ i -subgroup of G for every $$\sigma _{i}\in \Pi$$ σ i ∈ Π such that $$\sigma _i\cap \pi (G)\ne \emptyset$$ σ i ∩ π ( G ) ≠ ∅ . A subgroup H of G is called $$\Pi$$ Π -permutable or $$\Pi$$ Π -quasinormal in G if G possesses a complete Hall $$\Pi$$ Π -set $$\mathcal{H}$$ H such that $$AH^{x}=H^{x}A$$ A H x = H x A for all $$H\in \mathcal{H}$$ H ∈ H and $$x\in G$$ x ∈ G . We study the embedding properties of H under the hypothesis that H is $$\Pi$$ Π -permutable in G. Some well-known results are generalized.

### Journal

Monatshefte f�r MathematikSpringer Journals

Published: Dec 17, 2016

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations