The limit of transitivity of a substitution group
Marie J. Weiss · Transactions of the American Mathematical Society · 1930
6,499 409,799Bochert's inequality is of no interest for these small values of t.2. The first step in the study of the 2-ply transitive group G is the proof of the following lemma:Lemma.If t>5, the subgroup that fixes t letters of a t-ply transitive group fixes exactly t, t+l, or t+2 letters.Let Gx be the subgroup that fixes * letters of a /-ply transitive group G. Suppose that Gt fixes the t+r -l letters bx, b2, ■ ■ ■ , bt, ax, a2, ■ ■ ■, ar-i, r > 1.Now consider the largest subgroup of G<_i in which Gt is invariant.Its order is rgt, where gt is the order of Gt, and it has a regular constituent of degree r on the letters bx, ax, a2, ■ ■ ■ , a,-X.Since all the subgroups similar to Gt found in Gt~x are conjugate, all the subgroups similar to Gt in any of the preceding groups Gt-i, i = 2,3, ■ ■ ■ ,t,Go = G, form one complete set of conjugates.Then the largest subgroup of Gt-2 in which Gf is invariant has a doubly transitive constituent of degree r+1 on the letters b2, bx, ax, Oi, • • •,