Article
MSC:
06A06,
06A07,
06B20,
06B23,
20D25,
20D30,
20D40,
20E15,
20F22,
20K27 | MR 4801111 | Zbl 07953712 | DOI: 10.21136/MB.2023.0146-22
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Keywords:
alternating group; dicyclic group; pronormal subgroup; lattice of subgroups; lower semimodular lattice
alternating group; dicyclic group; pronormal subgroup; lattice of subgroups; lower semimodular lattice
Summary:
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.
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