On minimal degrees and base sizes of primitive permutation groups
Let G be a primitive permutation group of degree n, acting on a set \(\Omega\). Then the minimal degree \(\mu(G)\) is by definition the smallest number of points moved by any non-identity element of G. Before the completion of the classification of finite simple groups, it has been proved by \textit{L. Babai} [Ann. Math., II. Ser. 113, 553-568 (1981; Zbl 0485.20002)] that \(\mu(G)>(\sqrt{n}-1)/2\) provided that the alternating group \(A_ n\) is not contained in G. The author uses the classification of finite simple groups to improve this result. This is done via considering bases of G: A subset \(\Delta\) of \(\Omega\) is called a base of G is the pointwise stabilizer of \(\Delta\) in G is the identity; let b(G) denote the minimum size of any base in G. Babai proved that \(b(G)<4\sqrt{n}\log_ 2n\) [loc. cit.]. The paper's main results are the following: Theorem. One of the following holds: (i) G is a subgroup of \(S_ m wr S_ r\) containing \((A_ m)^ r\), of degree \(n=\binom{m}{k}^ r\); (ii) \(b(G)<9\log_ 2n.\) Corollary. If G is not in case (i) of the Theorem then \(\mu(G)>n/9\log_ 2n\). The examples \(AGL(d,2)\) of degree \(2^ d\) show that there exist primitive permutation groups G not satisfying (i) such that b(G) is roughly \(\log_ 2n\). The organization of the proof is canonical: reduction to the case of a simple socle and considering all possible cases in view of the classification. The corollary then follows from the well known fact that \(b(G)\mu(G)\geq n.\)
scientific article; zbMATH DE number 3865543
scientific article