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

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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 
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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

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