Subgroup-permutability and affine planes (Q5946244)

Let \(G\) be a group, \(E\) be the set of nonnormal subgroups of \(G\). Let \(\Gamma\) be a graph whose edge set is \(E\) and \(A,B\in E\) are adjacent if and only if \(AB=BA\). The authors study the case that \(\Gamma\) is the incidence graph of an affine plane. Then they prove that this plane is Desarguesian and \(G\) is a semidirect product of an elementary Abelian group \(V\) of order \(p^2\) and a cyclic group of order \(q^k\), \(k\geq 1\), which induces a group of power automorphisms of order \(q\) on \(V\).