%0 Journal Article
%T On the planarity of a graph related to the join of subgroups of a finite group
%J Bulletin of the Iranian Mathematical Society
%I Iranian Mathematical Society (IMS)
%Z 1017-060X
%A Taeri, B.
%A Ahmadi, H.
%D 2014
%\ 12/01/2014
%V 40
%N 6
%P 1413-1431
%! On the planarity of a graph related to the join of subgroups of a finite group
%K Graph on group
%K plannar graph
%K finite group
%R
%X Let $G$ be a finite group which is not a cyclic $p$-group, $p$ a prime number. We define an undirected simple graph $Delta(G)$ whose vertices are the proper subgroups of $G$, which are not contained in the Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge if and only if $G=langle H , Krangle$. In this paper we classify finite groups with planar graph. %For this, by Kuratowski's Theorem, we have to study subdivisions %of the Kuratowski graphs $K_{3 , 3}$ and $K_5$ in the graph $Delta(G)$. Our result shows that only few groups have planar graphs.
%U http://bims.iranjournals.ir/article_573_026d933a1762fba8b0e0f563507e5038.pdf