Permutation Modules with Nakayama Endomorphism Rings

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