# On the planarity of a graph related to the join of subgroups of a finite group

Document Type : Research Paper

Authors

Isfahan University of Technology

Abstract

‎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.

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### History

• Receive Date: 22 April 2013
• Revise Date: 03 September 2013
• Accept Date: 17 October 2013