Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X40620141201On the planarity of a graph related to the join of subgroups of a finite group14131431573ENB.TaeriIsfahan University of TechnologyH.AhmadiIsfahan University of TechnologyJournal Article20130422Let $G$ be a finite group which is not a cyclic $p$-group, $p$ a prime number. <br />We define an undirected simple graph $Delta(G)$ whose <br />vertices are the proper subgroups of $G$, which are not contained in the <br />Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge <br />if and only if $G=langle H , Krangle$. In this paper we classify finite groups with planar graph. <br />%For this, by Kuratowski's Theorem, we have to study subdivisions <br />%of the Kuratowski graphs $K_{3 , 3}$ and $K_5$ in the graph $Delta(G)$. <br />Our result shows that only few groups have planar graphs.http://bims.iranjournals.ir/article_573_026d933a1762fba8b0e0f563507e5038.pdf