On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

Abstract

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.

Keywords

• alternating group
• dicyclic group
• pronormal subgroup
• lattice of subgroups
• lower semimodular lattice