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On the intersection graph of a finite group

Hossein Shahsavari, Behrooz Khosravi (2017)

Czechoslovak Mathematical Journal

For a finite group G , the intersection graph of G which is denoted by Γ ( G ) is an undirected graph such that its vertices are all nontrivial proper subgroups of G and two distinct vertices H and K are adjacent when H K 1 . In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of Aut ( Γ ( G ) ) .

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

Shrawani Mitkari, Vilas Kharat (2024)

Mathematica Bohemica

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 L ( G ) and sublattices of L ( G ) . It is proved that the collections of all pronormal subgroups of A n and S n do not form sublattices of respective L ( A n ) and L ( S n ) , whereas the collection of all pronormal subgroups LPrN ( Dic n ) of a dicyclic group is a sublattice of L ( Dic n ) . Furthermore, it is shown that L ( Dic n ) and LPrN ( Dic n ) are lower semimodular lattices.

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