The stable subgroup graph

Abstract

In this paper we introduce stable subgroup graph associated to the group G. It is a graph with vertex set all subgroups of G and two distinct subgroupsThe planarity of the stable subgroup graph of solvable groups has been discussed. Finally, the induced subgraph of stable subgroup graph with vertex set whole non-normal subgroups is considered and its planarity is verified for some certain groups.

The Application of GAP Software in Constructing the Non-Normal Subgroup Graphs of Alternating Groups

“…Based on definition 3, the vertices of the non-normal subgroup graph are all the elements of 𝐴 4 and the direction of the edges is determined by the product of the two distinct elements. The following GAP coding is for one of the subgroups of the alternating group of order 12. gap> A4:=AlternatingGroup(4); Alt( [ 1 .. 4 ] ) gap> e:=Elements(A4); [ (), (2,3,4), (2,4,3), (1,2) (3,4), (1,2,3), (1,2,4), (1,3,2), (1,3,4), (1,3)(2,4), (1,4,2), (1,4,3), (1,4) Group([ (1,2)(3,4) ]) gap> Elements(H1); [ (), (1,2)(3,4) ] gap> for i in e do > for j in e do; > if i<>j then > c:=i*j; > d:=c in H1; > if (d=true) then > Print ("{",i,",",j,",",c,",",d,"},"); > fi; > fi; > od; > od; {(),(1,2) (3,4),(1,2)(3,4),true},{(2,3,4), (2,4,3),(),true},{ (2,3,4), (1,2 ,3),(1,2)(3,4),true},{(2,4,3), (2,3,4),(),true},{ (2,4,…”

“…These graphs were using the cyclic subgroups as vertices and two distinct subgroups are adjacent if one of them is a subset of the other. Then, more graphs were constructed using subgroups which are stable subgroup graph defined by Tolue [2] and normal subgroup-based power graph defined by Bhuniya and Sudip Bera [3]. Most of the research on graphs associated with the finite group were simple undirected graphs.…”