Geodesic
On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups
Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 427-438
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices ${\rm L}(G)$ and sublattices of ${\rm L}(G)$. It is proved that the collections of all pronormal subgroups of ${\rm A}_n$ and S$_n$ do not form sublattices of respective ${\rm L}({\rm A}_n)$ and ${\rm L}({\rm S}_n)$, whereas the collection of all pronormal subgroups ${\rm LPrN}({\rm Dic}_n)$ of a dicyclic group is a sublattice of ${\rm L}({\rm Dic}_n)$. Furthermore, it is shown that ${\rm L}({\rm Dic}_n)$ and ${\rm LPrN}({\rm Dic}_n$) are lower semimodular lattices.
In this paper, the structures of collection of pronormal subgroups of dicyclic, symmetric and alternating groups $G$ are studied in respect of formation of lattices ${\rm L}(G)$ and sublattices of ${\rm L}(G)$. It is proved that the collections of all pronormal subgroups of ${\rm A}_n$ and S$_n$ do not form sublattices of respective ${\rm L}({\rm A}_n)$ and ${\rm L}({\rm S}_n)$, whereas the collection of all pronormal subgroups ${\rm LPrN}({\rm Dic}_n)$ of a dicyclic group is a sublattice of ${\rm L}({\rm Dic}_n)$. Furthermore, it is shown that ${\rm L}({\rm Dic}_n)$ and ${\rm LPrN}({\rm Dic}_n$) are lower semimodular lattices.
DOI : 10.21136/MB.2023.0146-22
Classification : 06A06, 06A07, 06B20, 06B23, 20D25, 20D30, 20D40, 20E15, 20F22, 20K27
Keywords: alternating group; dicyclic group; pronormal subgroup; lattice of subgroups; lower semimodular lattice 
@article{10_21136_MB_2023_0146_22,
     author = {Mitkari, Shrawani and Kharat, Vilas},
     title = {On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups},
     journal = {Mathematica Bohemica},
     pages = {427--438},
     year = {2024},
     volume = {149},
     number = {3},
     doi = {10.21136/MB.2023.0146-22},
     mrnumber = {4801111},
     zbl = {07953712},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2023.0146-22/}
}
TY  - JOUR
AU  - Mitkari, Shrawani
AU  - Kharat, Vilas
TI  - On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups
JO  - Mathematica Bohemica
PY  - 2024
SP  - 427
EP  - 438
VL  - 149
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2023.0146-22/
DO  - 10.21136/MB.2023.0146-22
LA  - en
ID  - 10_21136_MB_2023_0146_22
ER  - 
%0 Journal Article
%A Mitkari, Shrawani
%A Kharat, Vilas
%T On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups
%J Mathematica Bohemica
%D 2024
%P 427-438
%V 149
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2023.0146-22/
%R 10.21136/MB.2023.0146-22
%G en
%F 10_21136_MB_2023_0146_22
Mitkari, Shrawani; Kharat, Vilas. On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups. Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 427-438. doi: 10.21136/MB.2023.0146-22

[1] Benesh, B.: A classification of certain maximal subgroups of alternating groups. Computational Group Theory and the Theory of Groups Contemporary Mathematics 470. AMS, Providence (2008), 21-26. | DOI | MR | JFM

[2] Călugăreanu, G.: Lattice Concepts of Module Theory. Kluwer Texts in the Mathematical Sciences 22. Kluwer, Dordrecht (2000). | DOI | MR | JFM

[3] Giovanni, F. de, Vincenzi, G.: Pronormality in infinite groups. Math. Proc. R. Ir. Acad. 100A (2000), 189-203. | MR | JFM

[4] Grätzer, G.: General Lattice Theory. Academic Press, New York (1978). | DOI | MR | JFM

[5] Hall, P.: Theorems like Sylow's. Proc. Lond. Math. Soc., III. Ser. 6 (1956), 286-304. | DOI | MR | JFM

[6] Lazorec, M.-S., Tărnăuceanu, M.: On some probabilistic aspects of (generalized) dicyclic groups. Quaest. Math. 44 (2021), 129-146. | DOI | MR | JFM

[7] Luthar, I. S.: Algebra. Volume 1. Groups. Narosa Publishing, New Delhi (1996). | JFM

[8] Mann, A.: A criterion for pronormality. J. Lond. Math. Soc. 44 (1969), 175-176. | DOI | MR | JFM

[9] Mitkari, S., Kharat, V., Agalave, M.: On the structure of pronormal subgroups of dihedral groups. J. Indian Math. Soc. (N.S) 90 (2023), 401-410. | MR | JFM

[10] Mitkari, S., Kharat, V., Ballal, S.: On some subgroup lattices of dihedral, alternating and symmetric groups. Discuss. Math., Gen. Algebra Appl. 43 (2023), 309-326. | DOI | MR | JFM

[11] Peng, T. A.: Pronormality in finite groups. J. Lond. Math. Soc., II. Ser. 3 (1971), 301-306. | DOI | MR | JFM

[12] Rose, J. S.: Finite soluble groups with pronormal system normalizers. Proc. Lond. Math. Soc., III. Ser. 17 (1967), 447-469. | DOI | MR | JFM

[13] Schmidt, R.: Subgroup Lattices of Groups. de Gruyter Expositions in Mathematics 14. Walter de Gruyter, Berlin (1994). | DOI | MR | JFM

[14] Stern, M.: Semimodular Lattices: Theory and Applications. Encyclopedia of Mathematics and Its Applications 73. Cambridge University Press, Cambridge (1999). | DOI | MR | JFM

[15] Suzuki, M.: Structure of a Group and the Structure of its Lattice of Subgroups. Springer, Heidelberg (1956). | DOI | MR | JFM

[16] Vdovin, E. P., Revin, D. O.: Pronormality and strong pronormality of subgroups. Algebra Logic 52 (2013), 15-23. | DOI | MR | JFM

Cité par Sources :