Planarity of the intersection graph of subgroups of a finite group. (Q2796993)

The intersection graph of a group \(G\) is the undirected graph whose vertices are the nontrivial proper subgroups of \(G\) and two vertices are joined by an edge if and only if they have a nontrivial intersection. Using Kuratowski's well-known theorem, the authors determine all finite groups with planar intersection graph. All these groups are structurally small: they are either cyclic of order \(p^i\), \(p^jq\), or \(pqr\) for pairwise different primes \(p,q,r\) and \(i\leq 5\), \(j\leq 2\), or are noncyclic of order \(p^2\) or \(4p\), or are certain semidirect products of a unique minimal normal subgroup of order \(p\) or \(p^2\) by a cyclic group of order \(q\), \(q^2\), or \(qr\).NEWLINENEWLINE We mention that \textit{S. Kayacan} and \textit{E. Yaraneri} proved the same result in their paper [J. Korean Math. Soc. 52, No. 1, 81-96 (2015; Zbl 1314.20016)].