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

Mitkari, Shrawani, and Kharat, Vilas. "On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups." Mathematica Bohemica 149.3 (2024): 427-438. <http://eudml.org/doc/299480>.

@article{Mitkari2024,
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.},
author = {Mitkari, Shrawani, Kharat, Vilas},
journal = {Mathematica Bohemica},
keywords = {alternating group; dicyclic group; pronormal subgroup; lattice of subgroups; lower semimodular lattice},
language = {eng},
number = {3},
pages = {427-438},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups},
url = {http://eudml.org/doc/299480},
volume = {149},
year = {2024},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 3
SP - 427
EP - 438
AB - 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.
LA - eng
KW - alternating group; dicyclic group; pronormal subgroup; lattice of subgroups; lower semimodular lattice
UR - http://eudml.org/doc/299480
ER -