The structure of the permutation modules for transitive \(p\)-groups of degree \(p^2\). (Q1109133)

Let \(G\) be a \(p\)-group acting transitively and faithfully on a set \(\Omega\) of size \(p^m\), let \(F\) be a field of characteristic \(p\). It is known that for the permutation module \(F\Omega\) the descending and ascending Loewy series coincide provided \(G\) has exponent \(p^m\) or \(G\) is Abelian. P. M. Neumann conjectured that this fact is true in general. The present nicely written paper contains a proof of Neumann's conjecture for the case \(m=2\). The proof of this result is inspired by the work of \textit{H. Wielandt} on transitive groups of degree \(p^2\) [Permutation groups through invariant relations and invariant functions (Lecture Notes, Ohio State Univ. 1969)].