Permutation Modules with Nakayama Endomorphism Rings

Given a field K of characteristic \(p>0\) and a natural number n, assuming that G is a permutation group acting on a set \(\Omega \) with n elements, then \(K\Omega \) is a permutation module for G in the natural way. If G is primitive and \(n\le 5p\), we will show that \(\textrm{End}_{KG}(K\Omega )\) is always a symmetric Nakayama algebra unless \(p=5\) and \(n=25\). As a consequence, \(\textrm{End}_{KG}(K\Omega )\) is always a symmetric Nakayama algebra if G is quasiprimitive, \(n<4p\) and \(3\not \mid p-1\) when \(n=3p\).