The Higman-McLaughlin theorem for flag-transitive linear spaces
A well known theorem by Higman and McLaughlin states that, if a group G acts flag-transitively on a finite linear space S, then G also acts point-primitively on S. The author proves the following generalization of that theorem. Given S and G as above, let \(G^ 0\) be a normal subgroup of G of index at most 3 (and assume S has at least two lines, to avoid degenerate situations). Then \(G^ 0\) is still point-primitive on S, apart from three exceptions when \([G:G^ 0]=3\) and \(S=AG(2,2)\), AG(3,4) or AG(2,8). The proof is an elegant example of interaction between geometric and group-theoretic methods and is readable for a non professional group theorist too (like the reviewer, for instance).
scientific article; zbMATH DE number 4171196