On minimal \(p\)-degrees in \(2\)-transitive permutation groups
If \(G\) is a transitive permutation group on a finite set \(\Omega\) of size \(n\) and if \(p\) is a prime divisor of \(|G|\), the minimal \(p\)-degree of \(G\), denote by \(m_p(G)\), is the smallest number of points moved by a non-identity \(p\)-element of \(G\). This is a generalization of the notion of minimal degree \(m(G)\) (the smallest number of points moved by any non-identity element), since \(m(G)=m_p(G)\) for a suitable prime divisor \(p\) of \(|G|\). Using the classification of finite 2-transitive permutation groups, the author shows that \(m_p(G)\geq\tfrac{p-1}{p+1}\cdot|\Omega|\) if \(G\) is 2-transitive and \(\text{Alt}(\Omega)\not\leq G\). All cases where \(m_p(G)<\tfrac{p-1}p\cdot|\Omega|\) are identified and minimal \(p\)-degrees are calculated for various 2-transitive permutation groups.
scientific article; zbMATH DE number 1333328