Simply transitive primitive groups

Abstract

1. The transitive constituents of the subgroup that leaves fixed one letter of a transitive group occur in pairs of equal degree. Transitive constituents on one letter are to be taken into account in the above statement. The two members of a pair sometimes coincide. This important property of transitive groups was proved by means of a certain quadratic invariant by Burnside in 1900.t It can be more easily demonstrated as follows: Let G be a non-regular transitive group and let the subgroup G1 of order gi that fixes one letter a of G have the transitive constituents B on the letters b, bi, * , bri-; C on the letters c, cl, , c,-1; . Consider a permutation S = (cab a ) a . Every one of the gi permutations G1S replaces a by b; and because c, cl, are the letters of a transitive constituent of G1, every permutation G1S is of the form (c'ab ) , where c' is some one of the letters c, cl, . Similarly every permutation SG, is of the form (cab' * ) . Then the array GlSGl includes every permutation of G in which a is preceded by one of the s cs or is folowed by one of the r bs. Now the number of distinct permutations in the array GlSGl is g12 divided by the number of permutations common to G1 and SGlS-l, or common to S-lG1S and G1,t that is, by the number of permutations of G1 that fix c or b. These numbers are gi/s and gl/r. Therefore r = s. If, as often happens, every permutation (ab ) is of the form (ab) or (blab . ). the transitive constituent B is paired with itself. Since the product