Covering monolithic groups with proper subgroups
Given a finite non-cyclic group $G$, call $sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $sigma(G) < sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is textit{monolith...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
University of Isfahan
2013-03-01
|
| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | http://www.theoryofgroups.ir/?_action=showPDF&article=2674&_ob=ea33a8a83df38a9fbe3ccb5bf483e473&fileName=full_text.pdf. |
| _version_ | 1828527913087008768 |
|---|---|
| author | Martino Garonzi |
| author_facet | Martino Garonzi |
| author_sort | Martino Garonzi |
| collection | DOAJ |
| description | Given a finite non-cyclic group $G$, call $sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $sigma(G) < sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups. |
| first_indexed | 2024-12-11T21:44:47Z |
| format | Article |
| id | doaj.art-c82c653995744f51a2682c3804de03a7 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-12-11T21:44:47Z |
| publishDate | 2013-03-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-c82c653995744f51a2682c3804de03a72022-12-22T00:49:41ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692013-03-0121131144Covering monolithic groups with proper subgroupsMartino GaronziGiven a finite non-cyclic group $G$, call $sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $sigma(G) < sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.http://www.theoryofgroups.ir/?_action=showPDF&article=2674&_ob=ea33a8a83df38a9fbe3ccb5bf483e473&fileName=full_text.pdf.CoversMonolithic groupsPrimitive groups |
| spellingShingle | Martino Garonzi Covering monolithic groups with proper subgroups International Journal of Group Theory Covers Monolithic groups Primitive groups |
| title | Covering monolithic groups with proper subgroups |
| title_full | Covering monolithic groups with proper subgroups |
| title_fullStr | Covering monolithic groups with proper subgroups |
| title_full_unstemmed | Covering monolithic groups with proper subgroups |
| title_short | Covering monolithic groups with proper subgroups |
| title_sort | covering monolithic groups with proper subgroups |
| topic | Covers Monolithic groups Primitive groups |
| url | http://www.theoryofgroups.ir/?_action=showPDF&article=2674&_ob=ea33a8a83df38a9fbe3ccb5bf483e473&fileName=full_text.pdf. |
| work_keys_str_mv | AT martinogaronzi coveringmonolithicgroupswithpropersubgroups |