Geodesic
On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups
Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 427-438

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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 
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
@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

[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 :