On the orders of primitive groups
The author tries to find bounds to the order of a primitive permutation group in function of its degree. His main theorem states that, with exceptions the four Mathieu groups, a primitive permutation group of degree \(n\) has order at most \[ n\cdot\prod^{[\log_2n]-1}_{i=0}(n-2^i)<n^{[\log_2n]+1} \] or has a socle isomorphic to a direct power of some alternating group. This bound is sharp. The proof of this theorem uses the classification of finite simple groups. The author establishes several corollaries to this result. He proves that a primitive permutation group of degree \(n\) has order less than \(n^{\sqrt n}\) if it is not 3-transitive and has order less than \(50\cdot n^{\sqrt n}\) if it does not contain \(A_n\). This is a sharp version of a result of \textit{P. J. Cameron} [Bull. Lond. Math. Soc. 13, 1-22 (1981; Zbl 0463.20003)]. He also improves a result of \textit{C. E. Praeger} and \textit{J. Saxl} [Bull. Lond. Math. Soc. 12, 303-307 (1980; Zbl 0443.20001)] by showing that a primitive permutation group of degree \(n\) not containing \(A_n\) has order less than \(3^n\), or even \(2^n\) if \(n>24\). The author also gives a list of the 24 primitive permutation groups of degree \(n\) whose order is greater than \(2^{n-1}\).
scientific article; zbMATH DE number 1868105
scientific article